Size: px
Start display at page:

Transcription

1 DIHEDRAL GROUPS II KEITH CONRAD We will characterize dihedral groups in terms of generators and relations, and describe the subgroups of D n, including the normal subgroups. We will also introduce an infinite group that resembles the dihedral groups and has all of them as quotient groups.. Abstract characterization of D n The group D n has two generators r and s with orders n and 2 such that srs r. We will show any group with a pair of generators like r and s (except for their order) admits a homomorphism onto it from D n, and is isomorphic to D n if it has the same size as D n. Theorem.. Let G be generated by elements a and b where a n for some n 3, b 2, and bab a. There is a surjective homomorphism D n G, and if G has order 2n then this homomorphism is an isomorphism. The hypotheses a n and b 2 do not mean a has order n and b has order 2, but only that their orders divide n and divide 2. For instance, the trivial group has the form a, b where a n, b 2, and bab a (take a and b to be the identity). When n 4, the group (Z/(8)) has generators 3 and 5 with 3 4, 5 2, and Proof. The equation bab a implies ba j b a j for any j Z (raise both sides to the jth power). Since b 2, we have for any k Z b k a j b k a ( )k j by considering even and odd k separately. Thus (.) b k a j a ( )kj b k. Thus, any product of a s and b s can have all the a s brought to the left and all the b s brought to the right. Taking into account that a n and b 2, we get (.2) G a, b {a j, a j b : j Z} {, a, a 2,..., a n, b, ab, a 2 b,..., a n b}. Thus G is a finite group with #G 2n. To write down an explicit homomorphism from D n onto G, the equations a n, b 2, and bab a suggest we should be able send r to a and s to b by a homomorphism. This suggests the function f : D n G defined by f(r j s k ) a j b k. This function makes sense, since the only ambiguity in writing an element of D n as r j s k is that j can change modulo n and k can change modulo 2, which has no effect on the right side since a n and b 2.

2 2 KEITH CONRAD and To check f is a homomorphism, we use (.): f(r j s k )f(r j s k ) a j b k a j b k a j a ( )k j b k b k a j+( )k j b k+k f((r j s k )(r j s k )) f(r j r ( )k j s k s k ) f(r j+( )k j s k+k ) a j+( )k j b k+k. The results agree, so f is a homomorphism from D n to G. It is onto since every element of G has the form a j b k and these are all values of f by the definition of f. If #G 2n then surjectivity of f implies injectivity, so f is an isomorphism. Remark.2. The homomorphism f : D n G constructed in the proof is the only one where f(r) a and f(s) b: if there is any such homomorphism then f(r j s k ) f(r) j f(s) k a j b k. So a more precise formulation of Theorem. is this: for any group G a, b where a n for some n 3, b 2, and bab a, there is a unique homomorphism D n G sending r to a and s to b. Mathematicians describe this state of affairs by saying D n with its generators r and s is universal as a group with two generators satisfying the three equations in Theorem.. As an application of Theorem., we can write down a matrix group over Z/(n) that is isomorphic to D n when n 3. Set {( ) } ± c (.3) Dn : c Z/(n) inside GL 2 (Z/(n)). The group D n has order 2n (since mod n for n 3). Inside D n, ( 0 ) has order 2 and ( ) has order n. A typical element of D n is ( ) ( ) ( ) ± c c ± 0 ( ) c ( ) ± 0, so ( ) and ( 0 ) generate D n. Moreover, ( ) and ( ) are conjugate by ( ( ) ( ) ( ) ( ) 0 ( ). ): Thus, by Theorem., Dn is isomorphic to D n, using ( ) in the role of r and ( 0 ) in the role of s. This realization of D n inside GL 2 (Z/(n)) should not be confused with the geometric cos(2π/n) sin(2π/n) realization of D n in GL 2 (R) using real matrices: r ( sin(2π/n) cos(2π/n) ) and s ( 0 0 ). Corollary.3. If n 6 is twice an odd number then D n Dn/2 Z/(2).

3 For example, D 6 D3 Z/(2) and D 0 D5 Z/(2). DIHEDRAL GROUPS II 3 Proof. Let H r 2, s, where r and s are taken from D n. Then (r 2 ) n/2, s 2, and sr 2 s r 2, so Theorem. tells us there is a surjective homomorphism D n/2 H. Since r 2 has order n/2, #H 2(n/2) n #D n/2, so D n/2 H. Set Z {, r n/2 }, the center of D n. The elements of H commute with the elements of Z, so the function f : H Z D n by f(h, z) hz is a homomorphism. Writing n 2k where k 2l + is odd, we get f((r 2 ) l, r n/2 ) r 2l+k r and f(s, ) s, so the image of f contains r, s D n. Thus f is surjective. Both H Z and D n have the same size, so f is injective too and thus is an isomorphism. There is no isomorphism as in Corollary.3 between D n and D n/2 Z/(2) when n is divisible by 4: since n and n/2 are even the center of D n has order 2 and the center of D n/2 Z/(2) has order Therefore the centers of D n and D n/2 Z/(2) are not isomorphic, so these groups are not isomorphic. 2. Dihedral groups and generating elements of order 2 In D n, we can obtain r from s and rs (just multiply: rs s rs 2 r), so we can use the reflections rs and s as generators for D n : D n r, s rs, s. In group-theoretic terms, D n is generated by two elements of order 2. They do not commute: rs s r and s rs srs r ss r. What finite groups besides D n for n 3 can be generated by two elements of order 2? Suppose G x, y, where x 2 and y 2. If x and y commute, then G {, x, y, xy}. This has size 4 provided x y. Then we see G behaves just like the group Z/(2) Z/(2), where x corresponds to (, 0) and y corresponds to (0, ). If x y, then G {, x} x is cyclic of size 2. If x and y do not commute, it turns out that G is essentially a dihedral group! Theorem 2.. Let G be a finite non-abelian group generated by two elements of order 2. Then G is isomorphic to a dihedral group. Proof. Let the two elements be x and y, so each has order 2 and G x, y. Since G is non-abelian and x and y generate G, x and y do not commute: xy yx. The product xy has some finite order, since we are told that G is a finite group. Let the order of xy be denoted n. Set a xy and b y. (If we secretly expect x is like rs and y is like s in D n, then this choice of a and b is understandable, since it makes a look like r and b look like s.) Then G x, y xy, y is generated by a and b, where a n and b 2. Since a has order n, n #G. Since b a, #G > n, so #G 2n. The order n of a is greater than 2. Indeed, if n 2 then a 2, so xyxy. Since x and y have order 2, we get xy y x yx, but x and y do not commute. Therefore n 3. Since (2.) bab yxyy yx, a y x yx, where the last equation is due to x and y having order 2, we obtain bab a. By Theorem., there is a surjective homomorphism D n G, so #G 2n. We saw before that #G 2n, so #G 2n and G D n.

4 4 KEITH CONRAD Theorem 2. says we know all the finite non-abelian groups generated by two elements of order 2. What about the finite abelian groups generated by two elements of order 2? We discussed this before Theorem 2.. Such a group is isomorphic to Z/(2) Z/(2) or (in the degenerate case that the two generators are the same element) to Z/(2). So we can define new dihedral groups D Z/(2), D 2 Z/(2) Z/(2). In terms of generators, D r, s where r and s has order 2, and D 2 r, s where r and s have order 2 and they commute. With these definitions, #D n 2n for every n, the dihedral groups are precisely the finite groups generated by two elements of order 2, the description of the commutators in D n for n > 2 (namely, they are the powers of r 2 ) is true for n (commutators are trivial in D and D 2, and so is r 2 in these cases), for even n, Corollary.3 is true when n is twice an odd number (including n 2) and false when n is a multiple of 4, the model for D n as a subgroup of GL 2 (R) when n 3 is valid for all n. However, D and D 2 don t satisfy all properties of dihedral groups when n > 2. For example, D n is non-abelian for n > 2 but not for n 2, the description of the center of D n when n > 2 (trivial for odd n and of order 2 for even n) is false when n 2, the matrix model for D n over Z/(n) doesn t work when n 2, Remark 2.2. Unlike finite groups generated by two elements of order 2, there is no elementary description of all finite groups generated by two elements with equal order > 2. Theorem 2.3. If N is a proper normal subgroup of D n then D n /N is a dihedral group. Therefore any nontrivial homomorphic image of a dihedral group is a dihedral group. Proof. The group D n /N is generated by rs and s, which both square to the identity, so they have order or 2 and they are not both trivial since D n /N is not trivial. If rs and s both have order 2 then D n /N is a dihedral group by Theorem 2. if D n /N if nonabelian or it is isomorphic to Z/(2) or Z/(2) Z/(2) if D n /N is abelian, which are also dihedral groups by our convention on the meaning of D and D 2. If rs or s have order then only one of them has order, which makes D n /N Z/(2) D. We will see what the proper normal subgroups of D n are in Theorem 3.8; aside from subgroups of index 2 (which are normal in any group) they turn out to be the subgroups of r. 3. Subgroups of D n We will list all subgroups of D n and then collect them into conjugacy classes of subgroups. Our results will be valid even for n and n 2. Recall D r, s where r and s has order 2, and D 2 r, s where r and s have order 2 and commute. Theorem 3.. Every subgroup of D n is cyclic or dihedral. A complete listing of the subgroups is as follows: () r d, where d n, with index 2d,

5 (2) r d, r i s, where d n and 0 i d, with index d. Every subgroup of D n occurs exactly once in this listing. DIHEDRAL GROUPS II 5 In this theorem, subgroups of the first type are cyclic and subgroups of the second type are dihedral: r d Z/(n/d) and r d, r i s D n/d. Proof. It is left to the reader to check n and n 2 separately. We now assume n 3. Let H be a subgroup of D n. The composite homomorphism H D n D n / r to a group of order 2 is either trivial or onto. Its kernel is H r. If the homomorphism is trivial then H H r, so H r, which means H r d for a unique d n. The order of r d is n/d and its index in D n is 2n/(n/d) 2d. If the homomorphism H D n / r is onto then H/(H r ) has order 2, so H r has index 2 in H. Set H r r d, so [H : r d ] 2. Since r d has order n/d, #H 2n/d and [D n : H] 2n/#H d. Choosing h H with h r d, we know h is not a power of r since r d H r, so h is a reflection. Write h r i s. Then H contains {r dk, r dk+i s : 0 k n d }, which is already 2(n/d) terms, so H r d, r i s. Multiplying r i s by an appropriate power of r d will produce an r j s where 0 j d, and we can replace r i s with this r j s in the generating set. So we may assume 0 i d. The subgroup r d, r i s is nontrivial and generated by two elements of order 2 (r i s and r d r i s), so it is isomorphic to a dihedral group. Since r d has order n/d, the order of r d, r i s is 2(n/d) 2n/d, whose index in D n is d. To check the two lists of subgroups in the theorem have no duplications, first we show the lists are disjoint. The only dihedral groups that are cyclic are groups of order 2, and r d, r i s has order 2 only when d n. The subgroup r n, r i s r i s has order 2 and r i s is not a power of r, so this subgroup is not on the first list. The first list of subgroups has no duplications since the order of r d changes when we change d (among positive divisors of n). If the second list of subgroups has a duplication, say r d, r i s r e, r j s, then computing the index in D n shows d e. The reflections in r d, r i s are all r dk+i s, so r j s r dk+i s for some k. Therefore j dk + i mod n, and from d n we further get j i mod d. That forces j i, since 0 i, j d. Corollary 3.2. Let n be odd and m 2n. If m is odd then there are m subgroups of D n with index m. If m is even then there is one subgroup of D n with index m. Let n be even and m 2n. If m is odd then there are m subgroups of D n with index m. If m is even and m doesn t divide n then there is one subgroup of D n with index m. If m is even and m n then there are m + subgroups of D n with index m. Proof. Check n and n 2 separately first. We now assume n 3. If n is odd then the odd divisors of 2n are the divisors of n and the even divisors of 2n are of the form 2d, where d n. From the list of subgroups of D n in Theorem 3., any subgroup with odd index is dihedral and any subgroup with even index is inside r. A subgroup with odd index m is r m, r i s for a unique i from 0 to m, so there are m such subgroups. The only subgroup with even index m is r m/2 by Theorem 3.. If n is even and m is an odd divisor of 2n, so m n, the subgroups of D n with index m are r m, r i s where 0 i m. When m is an even divisor of 2n, so (m/2) n, r m/2

6 6 KEITH CONRAD has index m. If m does not divide n then r m/2 is the only subgroup of index m. If m divides n then the other subgroups of index m are r m, r i s where 0 i m. From knowledge of all subgroups of D n we can count conjugacy classes of subgroups. Theorem 3.3. Let n be odd and m 2n. If m is odd then all m subgroups of D n with index m are conjugate to r m, s. If m is even then the only subgroup of D n with index m is r m/2. In particular, all subgroups of D n with the same index are conjugate to each other. Let n be even and m 2n. If m is odd then all m subgroups of D n with index m are conjugate to r m, s. If m is even and m doesn t divide n then the only subgroup of D n with index m is r m/2. If m is even and m n then any subgroup of D n with index m is r m/2 or is conjugate to exactly one of r m, s or r m, rs. In particular, the number of conjugacy classes of subgroups of D n with index m is when m is odd, when m is even and m doesn t divide n, and 3 when m is even and m n. Proof. As usual, check n and n 2 separately first. We now assume n 3. When n is odd and m is odd, m n and any subgroup of D n with index m is some r m, r i s. Since n is odd, r i s is conjugate to s in D n. The only conjugates of r m in D n are r ±m, and any conjugation sending s to r i s turns r m, s into r ±m, r i s r m, r i s. When n is odd and m is even, the only subgroup of D n with even index m is r m/2 by Theorem 3.. If n is even and m is an odd divisor of 2n, so m n, a subgroup of D n with index m is some r m, r i s where 0 i m. Since r i s is conjugate to s or rs (depending on the parity of i), and the only conjugates of r m are r ±m, r m, r i s is conjugate to r m, s or r m, rs. Note r m, s r m, r m s and r m s is conjugate to rs (because m is odd), Any conjugation sending r m s to rs turns r m, s into r m, rs. When m is an even divisor of 2n, so (m/2) n, Theorem 3. tells us r m/2 has index m. Any other subgroup of index m is r m, r i s for some i, and this occurs only when m n, in which case r m, r i s is conjugate to one of r m, s and r m, rs. It remains to show r m, s and r m, rs are nonconjugate subgroups of D n. Since m is even, the reflections in r m, s are of the form r i s with even i and the reflections in r m, rs are of the form r i s with odd i. Therefore no reflection in one of these subgroups has a conjugate in the other subgroup, so the two subgroups are not conjugate. Example 3.4. For odd prime p, the only subgroup of D p with index 2 is r and all p subgroups with index p (hence order 2) are conjugate to r p, s s. Example 3.5. In D 6, the subgroups of index 2 are r, r 2, s, and r 2, rs, which are nonconjugate to each other. All 3 subgroups of index 3 are conjugate to r 3, s. The only subgroup of index 4 is r 2. A subgroup of index 6 is r 3 or is conjugate to s or rs. Example 3.6. In D 0 the subgroups of index 2 are r, r 2, s, and r 2, rs, which are nonconjugate. The only subgroup of index 4 is r 2, all 5 subgroups with index 5 are conjugate to r 5, s, and a subgroup with index 0 is r 5 or is conjugate to r 0, s or r 0, rs. Example 3.7. When k 3, the dihedral group D 2 k has three conjugacy classes of subgroups with each index 2, 4,..., 2 k.

7 DIHEDRAL GROUPS II 7 We now classify the normal subgroups of D n, using a method that does not rely on our listing of all subgroups or all conjugacy classes of subgroups. Theorem 3.8. In D n, every subgroup of r is a normal subgroup of D n ; these are the subgroups r d for d n and have index 2d. This describes all proper normal subgroups of D n when n is odd, and the only additional proper normal subgroups when n is even are r 2, s and r 2, rs with index 2. In particular, there is at most one normal subgroup per index in D n except for three normal subgroups r, r 2, s, and r 2, rs of index 2 when n is even. Proof. We leave the cases n and n 2 to the reader, and take n 3. Since r is a cyclic normal subgroup of D n all of its subgroups are normal in D n, and by the structure of subgroups of cyclic groups these have the form r d where d n. It remains to find the proper normal subgroups of D n that are not inside r. Any subgroup of D n not in r must contain a reflection. First suppose n is odd. All the reflections in D n are conjugate, so a normal subgroup containing one reflection must contain all n reflections, which is half of D n. The subgroup also contains the identity, so its size is over half of the size of D n, and thus the subgroup is D n. So every proper normal subgroup of D n is contained in r. Next suppose n is even. The reflections in D n fall into two conjugacy classes of size n/2, represented by r and rs, so a proper normal subgroup N of D n containing a reflection will contain half the reflections or all the reflections. A proper subgroup of D n can t contain all the reflections, so N contains exactly n/2 reflections. Since N contains the identity, N > n/2, so [D n : N] < (2n)/(n/2) 4. A reflection in D n lying outside of N has order 2 in D n /N, so [D n : N] is even. Thus [D n : N] 2, and conversely every subgroup of index 2 is normal. Since D n /N has order 2 we have r 2 N. The subgroup r 2 in D n is normal with index 4, so the subgroups of index 2 in D n are obtained by taking the inverse image in D n of subgroups of index 2 in D n / r 2 {, r, s, rs} Z/(2) Z/(2): the inverse image of {, r} is r, the inverse image of {, s} is r 2, s, the inverse image of {, rs} is r 2, rs. Example 3.9. For an odd prime p, the only nontrivial proper normal subgroup of D p is r, with index 2. Example 3.0. In D 6, the normal subgroups of index 2 are r, r 2, s, and r 2, rs. The normal subgroup of index 4 is r 2 and of index 6 is r 3. There is no normal subgroup of index 3. Example 3.. The normal subgroups of D 0 of index 2 are r, r 2, s, and r 2, rs. The normal subgroup of index 4 is r 2 and of index 0 is r 5. There is no normal subgroup of index 5. Example 3.2. When k 3, the dihedral group D 2 k has one normal subgroup of each index except for three normal subgroups of index 2. The exceptional normal subgroups r 2, s and r 2, rs in D n for even n 4 can be realized as kernels of explicit homomorphisms D n Z/(2). In D n / r 2, s we have r 2 and s, so r a s b r a with a only mattering mod 2. In D n / r 2, rs we have r 2 and s r r, so r a s b r a+b, with the exponent only mattering mod 2. Therefore two

8 8 KEITH CONRAD homomorphisms D n Z/(2) are r a s b a mod 2 and r a s b a + b mod 2. These functions are well-defined since n is even and their respective kernels are r 2, s and r 2, rs. We can also see that these functions are homomorphisms using the general multiplication rule in D n : r a s b r c s d r a+( )bc s b+d. We have a + ( ) b c a + c mod 2 and a + ( ) b c + b + d (a + b) + (c + d) mod An infinite dihedral-like group In Theorem 2., the group is assumed to be finite. This finiteness is used in the proof to be sure that xy has a finite order. It is reasonable to ask if the finiteness assumption can be removed: after all, could a non-abelian group generated by two elements of order 2 really be infinite? Yes! In this appendix we construct such a group and show that there is only one such group up to isomorphism. Our group will be built out of the linear functions f(x) ax + b where a ± and b Z, with the group law being composition. For instance, the inverse of x is itself and the inverse of x + 5 is x 5. This group is called the affine group over Z and is denoted Aff(Z). The label affine is just a fancy name for linear function with a constant term. In linear algebra, the functions that are called linear all send 0 to 0, so ax + b is not linear in that sense (unless b 0). Calling a linear function affine avoids any confusion with the more restricted linear algebra sense of the term linear function. Since polynomials ax + b compose in the same way that the matrices ( a b ) multiply, we can consider such matrices, with a ± and b Z, as another model for the group Aff(Z). We will adopt this matrix model for the practical reason that it is simpler to write down products and powers with matrices rather than compositions with polynomials. Theorem 4.. The group Aff(Z) is generated by ( 0 ) and ( ). In the polynomial model for Aff(Z), the two generators in Theorem 4. are the functions x and x +. Proof. The elements of Aff(Z) have the form ( ) k (4.) or ( ) ( ) ( l l 0 (4.2) ( ) ) k ( ) l ( 0 While ( 0 ) has order 2, ( ) has infinite order. However, we can also generate Aff(Z) by two matrices of order 2. Corollary 4.2. The group Aff(Z) is generated by ( 0 0 ) and ( ), which each have order 2. In the polynomial model for Aff(Z), these generators are x and x. Proof. Check ( ) has order 2. By Theorem 4., it now suffices to show ( ) can be generated from ( 0 ) and ( ). It is their product (taken in the right order!): ( 0 )( ) ( ). ).

9 DIHEDRAL GROUPS II 9 Corollary 4.3. The matrices ( 0 0 ) and ( ) are not conjugate in Aff(Z) and do not commute with a common element of order 2 in Aff(Z). Proof. Any conjugate of ( 0 ) in Aff(Z) has the form ( 2b ) for b Z, and ( ) does not have this form. Thus, the matrices are not conjugate. In Aff(Z), ( 0 ) commutes only with the identity and itself. Corollary 4.2 shows Aff(Z) is an example of an infinite group generated by two elements of order 2. Are there other such groups, not isomorphic to Aff(Z)? No. Theorem 4.4. Any infinite group generated by two elements of order 2 is isomorphic to Aff(Z). Proof. Write such a group as G and its two generators of order 2 as x and y. Since G is infinite, x and y do not commute. (Otherwise x, y {, x, y, xy} has only 4 elements.) Since x x and y y, we do not need to use any exponents on x and y when writing products. The elements of G are strings of x s and y s, such as xyyxxyxyxyxyxyxxy. The relations x 2 and y 2 let us cancel any pair of adjacent x s or y s, so xyyxxyxyxyxyxyxxy can be simplified to xyxyxyxyx (xy) 4 x. Also, the inverse of such a string is again a string of x s and y s. As any element of G can be written as a product of alternating x s and y s, there are four kinds of elements, depending on the starting and ending letter: start with x and end with y, start with y and end with x, or start and end with the same letter. These four types of strings can be written as (4.3) (xy) k, (yx) k, (xy) k x, (yx) k y, where k is a non-negative integer. Before we look more closely at these products, let s indicate how the correspondence between G and Aff(Z) is going to work out. We want to think of x as ( 0 ) and y as ( ). Therefore the product xy should correspond to ( 0 )( ) ( ), and in particular have infinite order. Does xy really have infinite order? Yes, because if xy has finite order, the proof of Theorem 2. shows G x, y is a finite group. (The finiteness hypothesis on the group in the statement of Theorem 2. was only used in its proof to show xy has finite order; granting that xy has finite order, the rest of the proof of Theorem 2. shows x, y has to be a finite group.) The proof of Theorem 4. shows each element of Aff(Z) is ( )k or ( )k ( 0 ) for some k Z. This suggests we should show each element of G has the form (xy) k or (xy) k x. Let z xy, so z y x yx. Also xzx yx, so (4.4) xzx z. The elements in (4.3) have the form z k, z k, z k x, and z k y, where k 0. Therefore elements of the first and second type are just integral powers of z. Since z k y z k yxx z k x, elements of the third and fourth type are just integral powers of z multiplied on the right by x. Now we make a correspondence between Aff(Z) and G x, y, based on the formulas in (4.) and (4.2). Let f : Aff(Z) G by f ( k ) z k, f ( l ) z l x.

10 0 KEITH CONRAD This function is onto, since we showed each element of G is a power of z or a power of z multiplied on the right by x. The function f is one-to-one, since z has infinite order (and, in particular, no power of z is equal to x, which has order 2). By taking cases, the reader can check f(ab) f(a)f(b) for any A and B in Aff(Z). Some cases will need the relation xz n z n x, which follows from raising both sides of (4.4) to the n-th power. Remark 4.5. The abstract group x, y from this proof is the set of all words in x and y (like xyxyx) subject only to the relation that any pair of adjacent x s or adjacent y s can be cancelled (e.g., xyxxxy xyxy). Because the only relation imposed (beyond the group axioms) is that xx and yy are the identity, this group is called a free group on two elements of order 2. Corollary 4.6. Every nontrivial quotient group of Aff(Z) is isomorphic to Aff(Z) or to D n for some n. Proof. Since Aff(Z) is generated by two elements of order 2, any nontrivial quotient group of Aff(Z) is generated by two elements that have order or 2, and not both have order. If one of the generators has order then the quotient group is isomorphic to Z/(2) D. If both generators have order 2 then the quotient group is isomorphic to Aff(Z) if it is infinite, by Theorem 4.4, and it is isomorphic to some D n if it is finite since the finite groups generated by two elements of order 2 are the dihedral groups. Every dihedral group arises as a quotient of Aff(Z). For n 3, reducing matrix entries modulo n gives a homomorphism Aff(Z) GL 2 (Z/(n)) whose image is the matrix group D n from (.3), which is isomorphic to D n. The map ( a b ) (a, b mod 2) is a homomorphism from Aff(Z) onto {±} Z/(2) D 2 and the map ( a b ) a is a homomorphism from Aff(Z) onto {±} D. Considering the kernels of these homomorphisms for n 3, n 2, and n reveals that we can describe all of these maps onto dihedral groups in a uniform way: for any n, ( 0 n ) Aff(Z) and Aff(Z)/ ( n ) D n. This common pattern is another justification for our definition of the dihedral groups D and D 2.

WHY WORD PROBLEMS ARE HARD

WHY WORD PROBLEMS ARE HARD KEITH CONRAD 1. Introduction The title above is a joke. Many students in school hate word problems. We will discuss here a specific math question that happens to be named the

ISOMORPHISMS KEITH CONRAD 1. Introduction Groups that are not literally the same may be structurally the same. An example of this idea from high school math is the relation between multiplication and addition

CONSEQUENCES OF THE SYLOW THEOREMS

CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.

GENERALIZED QUATERNIONS

GENERALIZED QUATERNIONS KEITH CONRAD 1. introduction The quaternion group Q 8 is one of the two non-abelian groups of size 8 (up to isomorphism). The other one, D 4, can be constructed as a semi-direct

ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

5 Group theory. 5.1 Binary operations

5 Group theory This section is an introduction to abstract algebra. This is a very useful and important subject for those of you who will continue to study pure mathematics. 5.1 Binary operations 5.1.1

Algebra SEP Solutions

Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since

Yale University Department of Mathematics Math 350 Introduction to Abstract Algebra Fall Midterm Exam Review Solutions

Yale University Department of Mathematics Math 350 Introduction to Abstract Algebra Fall 2015 Midterm Exam Review Solutions Practice exam questions: 1. Let V 1 R 2 be the subset of all vectors whose slope

CYCLICITY OF (Z/(p))

CYCLICITY OF (Z/(p)) KEITH CONRAD 1. Introduction For each prime p, the group (Z/(p)) is cyclic. We will give seven proofs of this fundamental result. A common feature of the proofs that (Z/(p)) is cyclic

φ(xy) = (xy) n = x n y n = φ(x)φ(y)

Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =

Math 451, 01, Exam #2 Answer Key

Math 451, 01, Exam #2 Answer Key 1. (25 points): If the statement is always true, circle True and prove it. If the statement is never true, circle False and prove that it can never be true. If the statement

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

A. (Groups of order 8.) (a) Which of the five groups G (as specified in the question) have the following property: G has a normal subgroup N such that

MATH 402A - Solutions for the suggested problems. A. (Groups of order 8. (a Which of the five groups G (as specified in the question have the following property: G has a normal subgroup N such that N =

0 Sets and Induction. Sets

0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set

GROUPS OF ORDER p 3 KEITH CONRAD

GROUPS OF ORDER p 3 KEITH CONRAD For any prime p, we want to describe the groups of order p 3 up to isomorphism. From the cyclic decomposition of finite abelian groups, there are three abelian groups of

GALOIS GROUPS AS PERMUTATION GROUPS

GALOIS GROUPS AS PERMUTATION GROUPS KEITH CONRAD 1. Introduction A Galois group is a group of field automorphisms under composition. By looking at the effect of a Galois group on field generators we can

Algebraic structures I

MTH5100 Assignment 1-10 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one

Chapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples

Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter

Algebraic Structures Exam File Fall 2013 Exam #1

Algebraic Structures Exam File Fall 2013 Exam #1 1.) Find all four solutions to the equation x 4 + 16 = 0. Give your answers as complex numbers in standard form, a + bi. 2.) Do the following. a.) Write

GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

May 6, Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem. Show all of your work.

Math 236H May 6, 2008 Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem. Show all of your work. 1. (15 points) Prove that the symmetric group S 4 is generated

Math 370 Homework 2, Fall 2009

Math 370 Homework 2, Fall 2009 (1a) Prove that every natural number N is congurent to the sum of its decimal digits mod 9. PROOF: Let the decimal representation of N be n d n d 1... n 1 n 0 so that N =

Two subgroups and semi-direct products

Two subgroups and semi-direct products 1 First remarks Throughout, we shall keep the following notation: G is a group, written multiplicatively, and H and K are two subgroups of G. We define the subset

DISCRETE MATH (A LITTLE) & BASIC GROUP THEORY - PART 3/3. Contents

DISCRETE MATH (A LITTLE) & BASIC GROUP THEORY - PART 3/3 T.K.SUBRAHMONIAN MOOTHATHU Contents 1. Cayley s Theorem 1 2. The permutation group S n 2 3. Center of a group, and centralizers 4 4. Group actions

1 First Theme: Sums of Squares

I will try to organize the work of this semester around several classical questions. The first is, When is a prime p the sum of two squares? The question was raised by Fermat who gave the correct answer

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

ANALYSIS OF SMALL GROUPS

ANALYSIS OF SMALL GROUPS 1. Big Enough Subgroups are Normal Proposition 1.1. Let G be a finite group, and let q be the smallest prime divisor of G. Let N G be a subgroup of index q. Then N is a normal

GALOIS GROUPS OF CUBICS AND QUARTICS (NOT IN CHARACTERISTIC 2)

GALOIS GROUPS OF CUBICS AND QUARTICS (NOT IN CHARACTERISTIC 2) KEITH CONRAD We will describe a procedure for figuring out the Galois groups of separable irreducible polynomials in degrees 3 and 4 over

Groups and Symmetries

Groups and Symmetries Definition: Symmetry A symmetry of a shape is a rigid motion that takes vertices to vertices, edges to edges. Note: A rigid motion preserves angles and distances. Definition: Group

Modern Algebra I. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points.

1 2 3 style total Math 415 Please print your name: Answer Key 1 True/false Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1. Every group of order 6

GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur

Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Lecture - 05 Groups: Structure Theorem So, today we continue our discussion forward.

A Generalization of Wilson s Theorem

A Generalization of Wilson s Theorem R. Andrew Ohana June 3, 2009 Contents 1 Introduction 2 2 Background Algebra 2 2.1 Groups................................. 2 2.2 Rings.................................

SUPPLEMENTARY NOTES: CHAPTER 1

SUPPLEMENTARY NOTES: CHAPTER 1 1. Groups A group G is a set with single binary operation which takes two elements a, b G and produces a third, denoted ab and generally called their product. (Mathspeak:

NOTES ON FINITE FIELDS

NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining

Group Theory

Group Theory 2014 2015 Solutions to the exam of 4 November 2014 13 November 2014 Question 1 (a) For every number n in the set {1, 2,..., 2013} there is exactly one transposition (n n + 1) in σ, so σ is

CONJUGATION IN A GROUP

CONJUGATION IN A GROUP KEITH CONRAD 1. Introduction A reflection across one line in the plane is, geometrically, just like a reflection across any other line. That is, while reflections across two different

AN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS

AN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS The integers are the set 1. Groups, Rings, and Fields: Basic Examples Z := {..., 3, 2, 1, 0, 1, 2, 3,...}, and we can add, subtract, and multiply

FINITE GROUP THEORY: SOLUTIONS FALL MORNING 5. Stab G (l) =.

FINITE GROUP THEORY: SOLUTIONS TONY FENG These are hints/solutions/commentary on the problems. They are not a model for what to actually write on the quals. 1. 2010 FALL MORNING 5 (i) Note that G acts

Simple groups and the classification of finite groups

Simple groups and the classification of finite groups 1 Finite groups of small order How can we describe all finite groups? Before we address this question, let s write down a list of all the finite groups

Math 120 HW 9 Solutions

Math 120 HW 9 Solutions June 8, 2018 Question 1 Write down a ring homomorphism (no proof required) f from R = Z[ 11] = {a + b 11 a, b Z} to S = Z/35Z. The main difficulty is to find an element x Z/35Z

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,

Exercises on chapter 1

Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G

ENTRY GROUP THEORY. [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld.

ENTRY GROUP THEORY [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld Group theory [Group theory] is studies algebraic objects called groups.

and this makes M into an R-module by (1.2). 2

1. Modules Definition 1.1. Let R be a commutative ring. A module over R is set M together with a binary operation, denoted +, which makes M into an abelian group, with 0 as the identity element, together

Problem 1.1. Classify all groups of order 385 up to isomorphism.

Math 504: Modern Algebra, Fall Quarter 2017 Jarod Alper Midterm Solutions Problem 1.1. Classify all groups of order 385 up to isomorphism. Solution: Let G be a group of order 385. Factor 385 as 385 = 5

* 8 Groups, with Appendix containing Rings and Fields.

* 8 Groups, with Appendix containing Rings and Fields Binary Operations Definition We say that is a binary operation on a set S if, and only if, a, b, a b S Implicit in this definition is the idea that

2 Lecture 2: Logical statements and proof by contradiction Lecture 10: More on Permutations, Group Homomorphisms 31

Contents 1 Lecture 1: Introduction 2 2 Lecture 2: Logical statements and proof by contradiction 7 3 Lecture 3: Induction and Well-Ordering Principle 11 4 Lecture 4: Definition of a Group and examples 15

MA441: Algebraic Structures I. Lecture 26

MA441: Algebraic Structures I Lecture 26 10 December 2003 1 (page 179) Example 13: A 4 has no subgroup of order 6. BWOC, suppose H < A 4 has order 6. Then H A 4, since it has index 2. Thus A 4 /H has order

Definitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch

Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary

(d) Since we can think of isometries of a regular 2n-gon as invertible linear operators on R 2, we get a 2-dimensional representation of G for

Solutions to Homework #7 0. Prove that [S n, S n ] = A n for every n 2 (where A n is the alternating group). Solution: Since [f, g] = f 1 g 1 fg is an even permutation for all f, g S n and since A n is

Solutions to Assignment 4

1. Let G be a finite, abelian group written additively. Let x = g G g, and let G 2 be the subgroup of G defined by G 2 = {g G 2g = 0}. (a) Show that x = g G 2 g. (b) Show that x = 0 if G 2 = 2. If G 2

GROUP ACTIONS KEITH CONRAD. Introduction The groups S n, A n, and (for n 3) D n behave, by their definitions, as permutations on certain sets. The groups S n and A n both permute the set {, 2,..., n} and

Exercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups

Exercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups This Ark concerns the weeks No. (Mar ) and No. (Mar ). Plans until Eastern vacations: In the book the group theory included in the curriculum

ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS.

ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS. KEVIN MCGERTY. 1. RINGS The central characters of this course are algebraic objects known as rings. A ring is any mathematical structure where you can add

MATH 113 FINAL EXAM December 14, 2012

p.1 MATH 113 FINAL EXAM December 14, 2012 This exam has 9 problems on 18 pages, including this cover sheet. The only thing you may have out during the exam is one or more writing utensils. You have 180

Math 581 Problem Set 7 Solutions

Math 581 Problem Set 7 Solutions 1. Let f(x) Q[x] be a polynomial. A ring isomorphism φ : R R is called an automorphism. (a) Let φ : C C be a ring homomorphism so that φ(a) = a for all a Q. Prove that

GALOIS THEORY AT WORK: CONCRETE EXAMPLES

GALOIS THEORY AT WORK: CONCRETE EXAMPLES KEITH CONRAD 1. Examples Example 1.1. The field extension Q(, 3)/Q is Galois of degree 4, so its Galois group has order 4. The elements of the Galois group are

Definition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson

Definition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson On almost every Friday of the semester, we will have a brief quiz to make sure you have memorized the definitions encountered in our studies.

MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM

MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM Basic Questions 1. Compute the factor group Z 3 Z 9 / (1, 6). The subgroup generated by (1, 6) is

1 Chapter 6 - Exercise 1.8.cf

1 CHAPTER 6 - EXERCISE 1.8.CF 1 1 Chapter 6 - Exercise 1.8.cf Determine 1 The Class Equation of the dihedral group D 5. Note first that D 5 = 10 = 5 2. Hence every conjugacy class will have order 1, 2

TROPICAL SCHEME THEORY

TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),

Math 210A: Algebra, Homework 5

Math 210A: Algebra, Homework 5 Ian Coley November 5, 2013 Problem 1. Prove that two elements σ and τ in S n are conjugate if and only if type σ = type τ. Suppose first that σ and τ are cycles. Suppose

ABSTRACT ALGEBRA 1, LECTURES NOTES 5: SUBGROUPS, CONJUGACY, NORMALITY, QUOTIENT GROUPS, AND EXTENSIONS.

ABSTRACT ALGEBRA 1, LECTURES NOTES 5: SUBGROUPS, CONJUGACY, NORMALITY, QUOTIENT GROUPS, AND EXTENSIONS. ANDREW SALCH 1. Subgroups, conjugacy, normality. I think you already know what a subgroup is: Definition

Math 370 Spring 2016 Sample Midterm with Solutions

Math 370 Spring 2016 Sample Midterm with Solutions Contents 1 Problems 2 2 Solutions 5 1 1 Problems (1) Let A be a 3 3 matrix whose entries are real numbers such that A 2 = 0. Show that I 3 + A is invertible.

Rings If R is a commutative ring, a zero divisor is a nonzero element x such that xy = 0 for some nonzero element y R.

Rings 10-26-2008 A ring is an abelian group R with binary operation + ( addition ), together with a second binary operation ( multiplication ). Multiplication must be associative, and must distribute over

2) e = e G G such that if a G 0 =0 G G such that if a G e a = a e = a. 0 +a = a+0 = a.

Chapter 2 Groups Groups are the central objects of algebra. In later chapters we will define rings and modules and see that they are special cases of groups. Also ring homomorphisms and module homomorphisms

Teddy Einstein Math 4320

Teddy Einstein Math 4320 HW4 Solutions Problem 1: 2.92 An automorphism of a group G is an isomorphism G G. i. Prove that Aut G is a group under composition. Proof. Let f, g Aut G. Then f g is a bijective

Homework #5 Solutions

Homework #5 Solutions p 83, #16. In order to find a chain a 1 a 2 a n of subgroups of Z 240 with n as large as possible, we start at the top with a n = 1 so that a n = Z 240. In general, given a i we will

GROUPS AS GRAPHS. W. B. Vasantha Kandasamy Florentin Smarandache

GROUPS AS GRAPHS W. B. Vasantha Kandasamy Florentin Smarandache 009 GROUPS AS GRAPHS W. B. Vasantha Kandasamy e-mail: vasanthakandasamy@gmail.com web: http://mat.iitm.ac.in/~wbv www.vasantha.in Florentin

INTRODUCTION TO THE GROUP THEORY

Lecture Notes on Structure of Algebra INTRODUCTION TO THE GROUP THEORY By : Drs. Antonius Cahya Prihandoko, M.App.Sc e-mail: antoniuscp.fkip@unej.ac.id Mathematics Education Study Program Faculty of Teacher

3.1 Do the following problem together with those of Section 3.2:

Chapter 3 3.1 5. First check if the subset S is really a subring. The checklist for that is spelled out in Theorem 3.2. If S is a subring, try to find an element 1 S in S that acts as a multiplicative

Abstract Algebra II Groups ( )

Abstract Algebra II Groups ( ) Melchior Grützmann / melchiorgfreehostingcom/algebra October 15, 2012 Outline Group homomorphisms Free groups, free products, and presentations Free products ( ) Definition

Math 547, Exam 1 Information.

Math 547, Exam 1 Information. 2/10/10, LC 303B, 10:10-11:00. Exam 1 will be based on: Sections 5.1, 5.2, 5.3, 9.1; The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/547sp10/547.html)

(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d

The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers

EXAM 3 MAT 423 Modern Algebra I Fall c d a + c (b + d) d c ad + bc ac bd

EXAM 3 MAT 23 Modern Algebra I Fall 201 Name: Section: I All answers must include either supporting work or an explanation of your reasoning. MPORTANT: These elements are considered main part of the answer

Course 311: Abstract Algebra Academic year

Course 311: Abstract Algebra Academic year 2007-08 D. R. Wilkins Copyright c David R. Wilkins 1997 2007 Contents 1 Topics in Group Theory 1 1.1 Groups............................... 1 1.2 Examples of Groups.......................

ZORN S LEMMA AND SOME APPLICATIONS

ZORN S LEMMA AND SOME APPLICATIONS KEITH CONRAD 1. Introduction Zorn s lemma is a result in set theory that appears in proofs of some non-constructive existence theorems throughout mathematics. We will

FROM GROUPS TO GALOIS Amin Witno

WON Series in Discrete Mathematics and Modern Algebra Volume 6 FROM GROUPS TO GALOIS Amin Witno These notes 1 have been prepared for the students at Philadelphia University (Jordan) who are taking the

2a 2 4ac), provided there is an element r in our

MTH 310002 Test II Review Spring 2012 Absractions versus examples The purpose of abstraction is to reduce ideas to their essentials, uncluttered by the details of a specific situation Our lectures built

Computations/Applications

Computations/Applications 1. Find the inverse of x + 1 in the ring F 5 [x]/(x 3 1). Solution: We use the Euclidean Algorithm: x 3 1 (x + 1)(x + 4x + 1) + 3 (x + 1) 3(x + ) + 0. Thus 3 (x 3 1) + (x + 1)(4x

MATH 403 MIDTERM ANSWERS WINTER 2007

MAH 403 MIDERM ANSWERS WINER 2007 COMMON ERRORS (1) A subset S of a ring R is a subring provided that x±y and xy belong to S whenever x and y do. A lot of people only said that x + y and xy must belong

GALOIS THEORY AT WORK

GALOIS THEORY AT WORK KEITH CONRAD 1. Examples Example 1.1. The field extension Q(, 3)/Q is Galois of degree 4, so its Galois group has order 4. The elements of the Galois group are determined by their

Groups. 3.1 Definition of a Group. Introduction. Definition 3.1 Group

C H A P T E R t h r e E Groups Introduction Some of the standard topics in elementary group theory are treated in this chapter: subgroups, cyclic groups, isomorphisms, and homomorphisms. In the development

MODEL ANSWERS TO HWK #7. 1. Suppose that F is a field and that a and b are in F. Suppose that. Thus a = 0. It follows that F is an integral domain.

MODEL ANSWERS TO HWK #7 1. Suppose that F is a field and that a and b are in F. Suppose that a b = 0, and that b 0. Let c be the inverse of b. Multiplying the equation above by c on the left, we get 0

6 Permutations Very little of this section comes from PJE.

6 Permutations Very little of this section comes from PJE Definition A permutation (p147 of a set A is a bijection ρ : A A Notation If A = {a b c } and ρ is a permutation on A we can express the action

MAT 150A, Fall 2015 Practice problems for the final exam

MAT 150A, Fall 2015 Practice problems for the final exam 1. Let f : S n! G be any homomorphism (to some group G) suchthat f(1 2) = e. Provethatf(x) =e for all x. Solution: The kernel of f is a normal subgroup

Chapter 5. Modular arithmetic. 5.1 The modular ring

Chapter 5 Modular arithmetic 5.1 The modular ring Definition 5.1. Suppose n N and x, y Z. Then we say that x, y are equivalent modulo n, and we write x y mod n if n x y. It is evident that equivalence

Name: Solutions Final Exam

Instructions. Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [10 Points] All of

Some practice problems for midterm 2

Some practice problems for midterm 2 Kiumars Kaveh November 14, 2011 Problem: Let Z = {a G ax = xa, x G} be the center of a group G. Prove that Z is a normal subgroup of G. Solution: First we prove Z is

Introduction to Groups

Introduction to Groups Hong-Jian Lai August 2000 1. Basic Concepts and Facts (1.1) A semigroup is an ordered pair (G, ) where G is a nonempty set and is a binary operation on G satisfying: (G1) a (b c)

Number Axioms. P. Danziger. A Group is a set S together with a binary operation (*) on S, denoted a b such that for all a, b. a b S.

Appendix A Number Axioms P. Danziger 1 Number Axioms 1.1 Groups Definition 1 A Group is a set S together with a binary operation (*) on S, denoted a b such that for all a, b and c S 0. (Closure) 1. (Associativity)

Name: Solutions Final Exam

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [10 Points] For

ABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH

ABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH 1. Homomorphisms and isomorphisms between groups. Definition 1.1. Let G, H be groups.

Lecture 7 Cyclic groups and subgroups

Lecture 7 Cyclic groups and subgroups Review Types of groups we know Numbers: Z, Q, R, C, Q, R, C Matrices: (M n (F ), +), GL n (F ), where F = Q, R, or C. Modular groups: Z/nZ and (Z/nZ) Dihedral groups:

GROUP ACTIONS KEITH CONRAD 1. Introduction The symmetric groups S n, alternating groups A n, and (for n 3) dihedral groups D n behave, by their very definition, as permutations on certain sets. The groups