IIT Mumbai 2015 MA 419, Basic Algebra Tutorial Sheet-1 Let Σ be the set of all symmetries of the plane Π. 1. Give examples of s, t Σ such that st ts. 2. If s, t Σ agree on three non-collinear points, then show that s = t. 3. If t 1 Σ with t 2 = 1. Show that t is either a reflection or a rotation around some point of Π through angle π. 4. Prove that the composite of two reflections is a translation or a rotation according as the lines of reflection are parallel or not. Show conversely that every translation and every rotation can be expresses as a composite of two reflections. Therefore every symmetry is composition of atmost three reflections. 5. Prove that rotations and translations constitute a group of symmetries of the plane. 6. Prove that a glide reflection has no fixed points and that the inverse of a glide reflection is a glide reflection. If t is a translation through a non-zero vector V V not perpendicular to the line of reflection l of the reflection s, show that along with st, ts is also a glide reflection. What happens if l is perpendicular to V V? What is the square of a glide reflection? 7. (*) We say s, t Σ are conjugate if there exist u Σ such that s = u 1 tu. Show that this is an equivalence relation. Prove that conjugate symmetries have the same type, i.e. reflection, or both rotations, and so on. they are either both 8. Prove that the commutator [u, v] = u 1 v 1 uv of the symmetries u, v of the regular n-gon P is always a rotation around the center of P. 9. A symmetry u of the regular polygon P is called a central symmetry of P if it commutes with every symmetry of P. Show that u Σ(P ) is a central symmetry of P if and only if u commutes with the basic rotation s and with one reflection t in Σ(P ). Show that for odd n, 1 is the only central symmetry of P, whereas for even n, 1 and s n/2 are the only central symmetries of P.
10. Find all symmetries of (a) the single point, (b) a circle, (c) a parabola, (d) the configuration consisting of two points A, B, (e) area of the plane between two distinct parallel lines, (f) either of the two areas of the plane bounded by two distinct half rays with the same initial point. 11. If S is a semigroup, then S n is a semigroup with componentwise binary operation. Further projection map π i : S n S is a homomorphism. ( ) a b 12. Show that the set of matrices with a, b C is a semigroup, here a is b a complex conjugate of a. 13. If θ C satisfies aθ 2 + bθ + c = 0, a, b, c Z, a 0, show that Q[θ] := {x + yθ : x, y Q} is a multipicative semigroup with identity 1. Show that every non-zero element of this semigroup if a unit. 14. (*) Let ζ = e 2πi/n. Show that set A = {a 0 + a 1 ζ +... + a r ζ r : a i Z} is a multiplicative semigroup with identity 1. Show that A is same as with r n 1. Prove that elements ±ζ j, ± 1 ζk 1 ζ with (k, n) = 1 are units of the semigroup A. 15. Let u be a unit in semigroup S. If x S commutes with u k and u l with (k, l) = 1, then show that x commutes with u. 16. If φ : S S is surjective homomorphism of semigroups and if S has identity 1, show that φ(1) is identity of S. Give an example of homomorphism φ : S S of semigroups with identities 1, 1 such that φ(1) 1. 17. Determine all endomorphisms of additive semigroup Z. G is a group. 18. If G is even, show G contains x 1 with x 2 = 1. 19. If x 2 = 1 for all x G, then G is abelian. Give an example to show that x 3 = 1 for all x G does not implies that G is abelian. 20. If (xy) n = x n y n for three consecutive integers for all x, y G, then G is abelian. Give an example to show that it is not true if statement is given only for two consecutive integers. 21. Let p be a prime. Show that µ p = 1 µ p n is an abelian group under multiplication, here µ m is set of all m-th roots of 1 in C.
IIT Mumbai 2015 MA 419, Basic Algebra Tutorial Sheet-2 1. Show that the following set of matrices are groups under matrix multiplication. ( ) 1 n (i), n Z. 0 1 (ii) real non-singular upper triangular n n matrices. (iii) real non-singular upper triangular n n matrices with diagonal entries 1. ( ) ( ) ( ) ( ) 1 0 0 1 1 0 0 1 (iv),,,. 0 1 1 0 0 1 1 0 ( ) ( ) ( ) ( ) 1 0 1 0 0 1 0 1 (v) ±, ±, ±. ±. 0 1 0 1 1 0 1 0 ( a 2. Show that the matrices in SL 2 (Z) of the form c ) b with d (1) a = d = 1 mod n, b = c = 0 mod n; (2) c = 0 mod n; forms groups under multilication. ( ) a b 3. The complex matrix A = is unitary if and only if c d a 2 + b 2 = 1, c 2 + d 2 = 1, ac + bd = 0. ( Deduce that U 2 (C) consists of matrices A = α tβ Determine SU 2 (C). ( cos θ 4. Show that O 2 (R) consists of matrices sin θ Determine SO 2 (R) and show that it is commutative. ) β, α 2 + β 2 = 1, t = 1. tα ) ( sin θ cos θ, cos θ sin θ ) sin θ, θ real. cos θ 5. Let H denote the upper half plane consisting of complex numbers x + iy with y > 0. If a, b, c, d R with ad bc = 1, show that z w = az + b cz + d is a bijective mapping of H into itself. Show that the set of such transformations of H is a group under composition. Prove the same when we take a, b, c, d Z with ad bc = 1.
6. Determine all homomorphisms of Z, Z/n into (C, +), (C,.), (S 1,.). 7. If ζ = e 2πi/n, then show that ζ m = 1 if and only if n divides m; and µ n consists of 1, ζ,..., ζ n 1 ; and only relation satisfied by ζ is ζ n = 1. 8. Show that group (Z/n, +) is isomorphic to µ n. 9. Show that for any k 1, n 3, D n is homomorphic image of D kn. 10. Show that the group of automorphisms of the additive groups Z, Z/n, Q are isomorphic respectively to the multiplicative groups {±1}, (Z/n), Q. 11. Determine all homomorphisms from Q into R. 12. Show that additive groups Z, Q, R are non-isomorphic. What about R and C. Def. z G is central in group G if it commutes with all elements of G. 13. If z G commutes with set of elements generating G, then z is central in G. Find central elements in D n. 14. If x x n is an automorphism of G, show that for all x G, x n 1 is central in G. 15. Determine all central elements of GL 2 (Z), GL 2 (C), SL 2 (C). 16. Show that S 3 is isomorphic to D 3. 17. Show that the n 1 transpositions (12), (23),..., (n 1, n) generates S n. 18. Show that D 4 and Q 8 are not isomorphic. 19. Show that R and C are not isomorphic. 20. Find all automorphisms of D n.
IIT Mumbai 2015 MA 419, Basic Algebra Tutorial Sheet-3 1. Prove that S n, n 3 and A n, n 4 have trivial centers. 2. Find the center of D n. 3. Prove that [S n, S n ] = A n for n 1, [A n, A n ] = A n for n 5, and [A 4, A 4 ] = V 4. 4. Find the centralizer of (1 2 3 4 5) in S 5. 5. Show that the matrix I + ae pq for p q, a C is in SL n (C), where E pq is the matrix with 1 at (p, q) place and 0 elsewhere. By examining (I +ae pq )X = X(I +ae pq ) for p q, show that the center of GL n (C) consists of scalar matrices ai, a C. Find the center of SL n (C) and SL n (R). 6. Let X GL n (C). Prove that the operations of adding multiples of rows to other rows, X can ve reduced to the diagonal matrix D n (d) = diag(1,..., 1, d), d = det(x). Deduce that the matrices I + ae ij (a C, i j) generates SL n (C) and together with the matrices D n (d), d 0, they generate GL n (C). 7. Determine the orders of (a) a, 1, a ±1 in R. (b) 1, 3, 5, 7 in (Z/8Z). (c) 2 in (Z/19Z). (d) elements of S 3, S 4 and D n. ( ) ( ) 0 1 0 1 (e) s =, t =, u = st = 1 0 1 1 ( ) 1 1. 0 1 8. If a and b have finite orders in a group G, must ab also have finite order. Justify. 9. Show that (Z/19Z) is cyclic. 10. Show that for a, b, c in group G, ab and ba have same order. What about elements abc, bca, cab. 11. Prove that a k-cycle in S n has order k. Deduce that an element s S n has order the least common multiple of the lengths of its cycles. 12. If G has trivial center, show that inner automorphisms j a : G G defined by j a (x) = a 1 xa are distinct for all a G. Deduce that S 3 has atleast 6 distince automorphisms. Prove that any automorphism of S 3 permutes t = (12), t = (13), t = (23). Deduce that Aut(S 3 ) = S 3.
13. Let G be a cyclic group of order n (e.g. Z/nZ). Show that every element of G has order a divisor of n and for any divisor d of n, G contains exactly ϕ(d) elements of order d. Deduce the formula d/n ϕ(d) = n. 14. Find a system of representatives for G/H where G =< a > is cyclic or order n, and H =< a d >, d being a positive divisor of n. 15. Prove that the real numbers x, 0 x < 1 form a system of representatives for R/Z. 16. Determine [G : H] if finite, and a system of representatives for G/H when (a) G any group, H = G, H = {1}. (b) G = S 3, H =< (12) >, < (123) >. (c) G = (s, t s 2n = 1 = t 2, ts = s 1 t), H =< s 2, t >. (d) G = C, H = R, R +, T = S1. (f) G = GL n (C), H = SL n (C). (e) G = O n (R), H = SO n (R), (g) G = GL n (C), H = SL n (C). 17. If H, H are subgroups of G and xh = x H for some x, x G. Show that H = H. 18. If H, H are subgroups of G of finite index in G, show that so is H H. 19. If G is a finite group of order n and for each divisor d of n, G has atmost n elements. Show that G is cyclic. 20. If G is a finite abelian group of order n and k is an integer coprime to n, i.e. (n, k) = 1. Show that x x k is an automorphism of G.
IIT Mumbai 2015 MA 419, Basic Algebra Tutorial Sheet-4 1. Let G = µ p = n µ p n. Show that φ : G G defined by φ(x) = x p is a surjective group homomorphism. Conclude that G is isomorphic to its proper quotient group. 2. Show that Q/Z is isomorphic to µ p. 3. Using first isomorphism theorem, prove the following isomorphisms R/Z S 1, C/R R, C /R + T, GL n (C)/SL n (C) C, GL n (Z)/SL n (Z) {±1}. 4. Show that the set Inn(G) of inner automorphisms of G is a group under composition, isomorphic to G/Z(G). 5. Let G = D 8 =< s, t s 8 = t 2 = 1, ts = s 1 t > be the dihedral group of order 16. Show that subgroup < s 4 > is normal in G and let G = G/(s 4 ). (1) Find o(g); (2) Write elements of G as s a t b. (3) Show that H =< t, s 2 > is normal in G, and H is isomorphic to Klein 4-group. Find preimage of H in G. 6. Let a group G acts on a set A. Show that if b = g.a for a, b A and g G, then G b = gg a g 1. Find the kernel of the action if G acts transitively on A. 7. Let G be a permutation group on A, i.e. G S A. For a A, σ G, show that σg a σ 1 = G σ(a). Show that if G acts transitively of A, then σ G σg a σ 1 = 1. 8. Assume G S A is abelian and acts transitively on A. Show that σ(a) a for any a A if σ 1. Deduce that G = A. 9. Represent Klein 4 group G as a subgroup of S 4 under a left regular representation. 10. Use left regular representation of Q 8 to produce two elements of S 8 which generate a subgroup of S 8 isomorphic to Q 8. 11. If H, K are normal subgroups of G with H K = (1), show that every element of H commutes with every element of K. 12. Show that for any subgroup H of G, x G x 1 Hx is a normal subgroup of G. 13. Let G 2 denote the subgroup of G generated by the squares of all elements of G. Show that G 2 is normal in G and G/G 2 is abelian. Conclude that every commutator is a product of squares. 14. Show that a normal subgroup H of G of finite index [G : H] contains all elements of G of order prime to the index [G : H].
15. If N is a normal subgroup of G and H a characteristic subgroup of N (i.e. σ(h) = H for all σ Aut(N)). Show that H is normal in G. 16. If H is a fully invariant subgroup of G (i.e. σ(h) H for all σ End(G)), and L a fully invariant subgroup of H, show that L is a fully invariant subgroup of G. 17. If H is a characteristic subgroup of G, and L a characteristic subgroup of H, show that L is a characteristic subgroup of G. 18. Show that the transformations of the upper half plane H = {x + iy x, y R, y > 0 of the form z z = az + b, a, b, c, d Z with ad bc = 1, form a group under cz + d composition, isomorphic to SL 2 (Z)/{±Id}. Deduce that this group is generated by the transformations z 1/z and z z+1. 19. If G is a group of order n and p a prime which is the smallest divisor of n; if G has a conjugacy class of cardinality p, then G has a normal subgroup of index p in G. 20. Enumerate the elements of the conjugacy classes of each of D 4, Q 8, S n, A n for n 6. 21. Find the conjugates in S 3 of its subgroups < (23) > and < (123) >. Find conjugates of V 4 in S 4, of A n in S n, and of the subgroup of rotations in D n. 22. If H is a subgroup of G with [G : H] finite. Show that the number of conjugates of H in G is a divisor of [G : H]; so it is finite. 23. If G is a finite group and H G a proper subgroup. Show that G can not be the union of conjugates of H. 24. Another proof of Cauchy s theorem: Suppose p is a prime divisor of order n of finite group G. Let X be the set of all p-tuples (x 1,..., x p ) of elements of G such that x 1... x p = 1. Show that (i) If (x 1,..., x p ) X, then (x 2,..., x p, x 1 ) X. Hence the cyclic group C =< s > of order p acts on X as s.(x 1,..., x p ) = (x 2,..., x p, x 1 ). Each orbit has cardinality 1 or p. (ii) The orbit of (x 1,..., x p ) consists of only one element if and only if x 1 = x 2 =... = x p. (iii) Show that X = G p 1. Deduce there is an orbit other than {(1,..., 1)} having only one element in it. (iv) Show that there exists x 1 in G such that x p = 1.