Abstract Algebra II Groups ( )


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1 Abstract Algebra II Groups ( ) Melchior Grützmann / melchiorgfreehostingcom/algebra October 15, 2012
2 Outline Group homomorphisms Free groups, free products, and presentations Free products ( )
3 Definition and Examples Normal subgroups ( ) III Subgroups ( ) and homomorphisms ( ) Example Given again the subgroup A 3 S 3, we have already proved that it is a normal subgroup The two cosets are A 3 and S 3 \ A 3 = (12)A 3 Therefore the group structure on S 3 /A 3 is just ({±1}, ) Corollary Given a normal subgroup N G, then the subgroups of G/N are in 1:1correspondence with subgroups {S G : N S} Proof Let us denote π : G G/N the projection Given a subgroup S G, then it projects onto a subgroup π(s) G/N Conversely, given S G/N then it comes from a subgroup π 1 (S ) G that contains N = π 1 (id ) To see that this is the only subgroup containing N and projecting onto S, note that π(g) S is equivalent to gn S But then all of gn must be contained in the subgroup S G that projects onto S
4 Definition and Examples Subgroups ( ) and homomorphisms ( ) Normal subgroups ( ) IV Corollary Given a normal subgroup N G, then π and π 1 map inclusions to inclusions (ie N S 1 S 2 G, then π(s 1 ) π(s 2 ) and the analogue for π 1 ) and normal subgroups onto normal subgroups (ie K G implies π(k) G/N) The proof of preservation of normality is left as an exercise
5 Definition and Examples Subgroups ( ) and homomorphisms ( ) Center, Centralizer, and Normalizer In the search for normal subgroups we also obtain the following two notions: Definition Given a group G, then we define 1 The center ( ) of G as cent G := {z G : g G : zg = gz}, 2 Given an abelian subgroup H G the centralizer ( ) cent G H := {g G : h H : gh = hg}, 3 Given a subgroup H G its normalizer ( ) N G (H) := {g G : gh = Hg}
6 Definition and Examples Subgroups ( ) and homomorphisms ( ) Center, Centralizer, and Normalizer II Example Consider the group D 4 Its center is trivial, ie {id} as the multiplication table shows Consider further its subgroup H = σ A = {id, σ A } It is not normal, because σ a H Hσ a, but it is abelian Its centralizer is cent D4 H = σ A, σ B, τ 2 and its normalizer is also N S4 (H) = σ A, σ B, τ 2 Remark Given a group G, then its center is a normal abelian subgroup Given an abelian subgroup H G, then its centralizer is the biggest subgroup of elements commuting with all elements of H (In particular H, cent(g) cent G (H)) Given any subgroup H G, then its normalizer is the biggest subgroup in which H is a normal subgroup
7 Definition and Examples Subgroups ( ) and homomorphisms ( ) Endomorphisms ( ) and Automorphisms ( ) Definition Given a group G, then the endomorphisms ( ) End(G) are the homomorphism of G into itself The automorphisms ( ) Aut(G) are the isomorphisms of G onto itself Note that the endomorphisms form a monoid while the automorphisms form a group Example Consider C 3 = Z/(3) = [1] Obviously any endomorphism is specified by the image of [1] The endomorphisms are therefore {[0], [1], [2]} with the operation (multiplication), the identity Id C3 = [1] The automorphisms are those endomorphisms that are invertible, ie Aut(C 3 ) = {[1], [2]} = End(C 3 ) under the same operation Summary: The study of groups means the study of group structure together with the subgroup structure, endo, iso, and other homomorphisms (to and from groups)
8 Definition and Examples Subgroups ( ) and homomorphisms ( ) Exercises V Exercise Prove that the union of an increasing sequence of normal subgroups N 1 N 2 N 3 G, N i G of a group G is normal i N i G Exercise a Let G be a group generated by X G Prove that for two homomorphisms ϕ, ψ : G H into any group H, ϕ(x) = ψ(x) for all x X is equivalent to ϕ = ψ b Find all endomorphisms of V 4 := (12)(34), (13)(24), (14)(23) S 4 (Klein s four group) c Find all automorphisms of V 4 d Find all endomorphisms and automorphisms for D 3
9 23 The quotient map G G/N of a group G by a normal subgroup N G gives an example of a universal map ( ) in the following way Proposition Let N G be a normal subgroup Then every homomorphism ϕ: G H whose kernel contains N ker ϕ factors uniquely through the quotient map π : G G/N, ie there is a unique group homomorphism ϕ: G/N H such that ϕ = ϕ π Proof Idea ϕ(gn) := ϕ(g), but first check representation independence Let thus gn = hn for some g, h G But then ϕ(g) = ϕ(gn) = ϕ(hn) = ϕ(h) and thus ϕ is well defined The homomorphism properties follow now from those of ϕ The last property follows from the definition of ϕ
10 Theorem (First isomorphism theorem) Given a group homomorphism ϕ: G H, then G/ ker ϕ = im ϕ Proof The obvious candidate for the isomorphism is ϕ: G/ ker ϕ im ϕ : g ker ϕ ϕ(g) First let us check that ϕ is well defined Let thus N := ker ϕ and gn = hn for some g, h G But then ϕ(g) = ϕ(gn) = ϕ(hn) = ϕ(h) and thus the image is the same Second, note that gn ker ϕ implies ϕ(gn) = ϕ(g) = id H and thus g N Therefore ϕ is injective Finally note that every h im ϕ has a g G with ϕ(g) = h But then ϕ(gn) = h and thus ϕ is also surjective and therefore bijective, ie an isomorphism
11 Application: cyclic groups I Example Let g G be an element, then g is cyclic Consider the map ϕ: Z G : n g n If m > n N with g m = g n, then g m n = id Let now N be the smallest such difference Then g m = id for every N m and thus ϕ: Z/(N) g is welldefined, maps 1 g and thus also surjective Moreover ker ϕ = Z/(N) and thus ϕ is also injective, thus an isomorphism If there is no m > n N with g m = g n, then all the g m are disjoint, moreover they are also disjoint from g m for any m N Thus ϕ: Z g is a homomorphism, surjective, and also injective Therefore Z = g In particular every two cyclic groups of order n are isomorphic We denote by C n the cyclic group of order n
12 Subgroups of cyclic groups Proposition For a cyclic group of order n there is for every divisor d n a unique subgroup of order d Proof Let C n = g be a cyclic group and g of order n Define S d := g n/d Clearly S d is a cyclic group which moreover has d elements, because for h 0 := g n/d, h d 0 = id with 0 d < d would imply that h has order less than d and thus g order less than n which contradicts the assumption Conversely it is also possible to define a set S d := {h G : hd = id}, ie all those elements for which d is an exponent But since g = G for every h S d there is an m N such that h = gm h d = id implies dm 0 (mod n), ie dm = kn for some k N But then m = k n d and thus h = hk 0, ie h S d and thus S d S d The other inclusion is obvious, and thus S d = S d, ie both definitions coincide and the subgroup S d is unique (ie independent of g G as long as g = G)
13 Theorem (Second isomorphism theorem) Let G be a group and N, K G be normal subgroups If K N, then K N, N/K G/K and (G/K)/(N/K) = G/N Idea: K N G induce the following maps G π G/K ρ τ σ G/N (G/K)/(G/N) θ
14 Proof Let π : G G/K and ρ: G G/N be the quotient maps We show that there is a unique isomorphism θ : G/N (G/K)/(N/K) such that θ ρ = τ π where we also need to show that there is a morphism τ : G/K (G/K)/(N/K) First note that ρ factors through π, because K N, ie there is some homomorphism σ : G/K G/N : gk gn such that ρ = σ π Since ρ is surjective, so is σ We show that ker σ = N/K First note that K N, because K G For n N we have σ(nk) = nn = N Conversely if σ(gk) = N, then gn = N and thus g N This shows ker σ = N/K Therefore in particular N/K G/K Now the first isomorphism theorem yields an isomorphism θ : G/N (G/K)/(N/K) such that θ σ = τ Then θ ρ = τ π Since ρ is surjective, θ is unique with this property This completes the proof
15 Proposition (Third isomorphism theorem) Given a group G together with a subgroup S G and a normal subgroup N G, then SN G is a subgroup, S N S is a normal subgroup of S, and S/(S N) = (SN)/N Subgroup inclusion pattern: The required maps are: G SN ı N S N (SN)/N π θ SN S/(S N) ρ ı S S Ie the arrows to the downleft indicate normal subgroups, the others are just subgroups, and the groups to the right (along an arrow) are subgroups of the groups to the left
16 Proof First note that SN G is indeed a group, because N is normal Moreover N (SN) We will show that there is a unique isomorphism θ : S/(S N) (SN)/N such that θ ρ = π ı where π : SN (SN)/N is the projection, ı: S SN the inclusion, and there is a surjective homomorphism ρ: S S/(S N) Let ϕ: S (SN)/N : s sn, ie ϕ = π ı and ϕ is surjective Moreover ϕ(g) = N iff g N, ie ker ϕ = S N which is therefore normal in S Therefore ρ is just the canonical projection (and in particular surjective) Again by the first isomorphism theorem (SN)/N = im ϕ = S/ ker ϕ = S/(S N), ie there is an isomophism θ : S/(S N) (SN)/N Moreover the isomorphism constructed in the proof of the theorem fulfills θ ρ = ϕ = π ı and is thus unique, because ρ is surjective This completes the proof
17 Application: This implies in particular that the intersection of two normal subgroups of finite index has finite index
18 Exercises I Exercise Let ϕ: A B and ψ : A C be group homomorphisms Prove the following: If ψ is surjective, then ϕ factors through ψ if and only if ker ψ ker ϕ In this case ϕ factors uniquely through ψ Exercise Show that the identity homomorphism Id: 2Z 2Z does not factor through the inclusion homomorphism ı: 2Z Z even though ker ı ker Id Hint: Opposite to the situation in Exercise 31, ı is not surjective Exercise Let ϕ: A C and ψ : B C be group homomorphisms Prove the following: If ψ is injective, then ϕ factors through ψ if and only if im ϕ im ψ In this case ϕ factors uniquely through ψ
19 Exercises II Exercise Show that every subgroup of a cyclic group is cyclic Exercise a Show that the additive group R/Z is isomorphic to the multiplicative group of all complex numbers C of modulus 1 b Show that the additive group Q/Z is isomorphic to the group of all complex roots of unity (ie all complex numbers z 0 such that z is finite in C ) c Show that the complex nth roots of unity Ω n := {z C : z n = 1} form a cyclic group (wrt multiplication)
20 Exercises III Exercise Consider the group D 4 := σ, τ : σ 2 = id = τ 4, στσ = τ 1 a Find the order of every element in D 4, b Show that for every d (D 4 : 1) there is a subgroup S D 4 of order d Exercise a Let G be a finite group and S, T G any subgroups Show that ST = S T / S T b Find a group G together with subgroups S, T G such that ST G is not a group
21 Exercises IV Exercise Let G be a finite group, N G a normal subgroup and H G any subgroup such that N and (G : N) are relatively prime Show that H N iff H divides N Hint: Consider HN G
22 Free groups ( ), free products ( ), and presentations ( ) Example Consider the group Z/(4) it can be generated by the element [1] (as well as [3]), because [1] + [1] = [2], [1] + [2] = [3], [1] + [3] = [0] Moreover the elements fulfill the trivial relations a + (b + c) = (a + b) + c, [0] + a = a = a + [0], as well as b + a = a + b, 4a := a + a + a + a = 0 and infinitely many more Definition A presentation of a group is a set S of generators together with a set R of relations between them Notation G = s S : R where we understand associativity and the role of the neutral element and inverse elements to be implied
23 Lemma ( ) Free groups ( ) Given a finite set S of generators there is a group F(S) that is generated by S and for every map ϕ: S G into a group G there is a unique group homomorphism ϕ: F(S) G Example 1 These groups are called free groups and the simplest example of a free group is F 1 = Z generated by 1 This is a cyclic group 2 The free group on the generators S is (isomorphic to) the set of all (finite) cancelled words in S S A word abc z is called cancelled if there are no adjacent aā or āa for any a S The group operation is concatenation together with canceling, ie aā ϵ, āa ϵ for all a S
24 Theorem Given a set S of generators and a set of relations ( ) R in elements of S, then there is a group generated by S that fulfills only the relations R S Sketch of the proof The idea is to start from the free group F(S) and to impose the relations R In order to do that we consider the normal subgroup R S F(S) that is generated by R The quotient is clearly a group generated by the images of S Example 1 The dihedral group is D n := σ, τ : σ 2 = id = τ n, στσ = τ 1 2 The cyclic groups Z/(n) = 1 : n 1 = 0
25 Quaternions The quaternions ( ) are defined as H := R(i, j, k) where the unit quaternions i, j, and k multiply as i 2 = 1 = j 2 = k 2 and ij = k = ji, jk = i = kj, ki = j = ik These units form a group Q = {±1, ±i, ±j, ±k} Obviously Q is generated by i and j, also i 4 = 1 and i 2 = j 2 But we also have the relation jij 1 = i 1 We want to show that these three generate all relations in Q, ie that Q = i, j : i 4 = id, i 2 = j 2, jij 1 = i 1 Denote the second group by Q and note that every element in Q can be written as a finite sequence of is and js Combining adjacent is to i m and js to j n, we see that 0 m, n 3 Moreover the last relation (ji = i 1 j) permits us to move every i to the left of all j Therefore we are left with at most 16 elements But we also see that we can replace j 2 by i 2 and thus j 3 by i 2 j Therefore we are left with 8 elements which are exactly the elements of Q and thus Q = Q
26 Corollary If G is a group generated by a subset S G, then there is a unique surjective homomorphism ϕ: F(S) G that is the identity on S
27 Corollary (Free product ) Given two groups G and H where we assume G H = {id} there is a unique group G H that has G H as generators and fulfills only the relations R G, R H G H Example Given two free groups F m and F n then their free product F m F n = Fm+n is another free group Remark A bit more difficult to show is that a subgroup of a free group is again a free group (possibly in 0 generators)
28 amalgamation ( ) G H = S a joint subgroup Denote γ : G G H and θ : H G H the inclusions into the free product Then G S H = G H/ γ(s)θ(s 1 ) : s S G H, ie the amalgamation of G and H over S is the quotient of the free product by the normal subgroup spanned by the antidiagonal embedding of the intersection It can be shown for γ : G G S H and θ : H G S H that G S H is generated by γ(g) θ(h) and im γ im θ = γ(s) = θ(s) Analogously to the free product, the amalgamated product fulfills the universality property: For every pair of group homomorphisms ϕ G : G U and ϕ H : H U such that S = G H and ϕ G (S) = ϕ H (S), there is a unique ϕ: G S H U such that ϕ G = ϕ γ and ϕ H = ϕ θ
29 Exercise Given a group G, the conjugates of an element x G are C x := {gxg 1 : g G} Given a subset S G, there exists a smallest normal subgroup N G that contains S N Show that N consists of all products of elements in C S S 1 Exercise a List (compactly) all elements of the group a, b : a 2 = id = b 2 Give a compact multiplication table of the group b List all elements of the group a, b : a 2 = id = b 2 = (ab) 3 and give their multiplication table Which known group is it isomorphic to?
30 Exercise The multiplication of the unit quaternions i 2 = 1 = j 2 = k 2, ij = k = ji, jk = i = kj, ki = j = ik together with Rlinearity implies for a, b, c, d, a, b, c, d R, (a + bi + cj + dk)(a + b i + c j + d k) = =(aa bb cc dd ) + (ab + ba + cd dc )i + (ac + ca + db bd )j + (ad + da + bc cb )k a Show that the multiplication is associative b Let a + bi + cj + dk := a bi cj dk and z 2 := z z for every quaternion z H Show that z 1 z 2 = z 1 z 2 for every pair of quaternions z 1/2 H c Conclude that H \ {0} is a group under multiplication (What is the inverse? Therefore H is called a division algebra)
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