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Math 250A Homework 1
1) Let G be a group. A commutator in G is an element of the form aba−1b −1 with a,b ∈ G. Let G c be the subgroup generated by the commutators, called the commutator subgroup. Show that G c is normal in G. Challenge: Show that any homomorphism φ of G into an abelian group factors through G/G c , meaning that there exists a map f such that φ = f ◦π where π : G/G c is the canonical morphism.
2) Two subgroups H and H 0 of a group G are said to be commensurable if H ∩H 0 is of finite index in both H and H 0 . Show that commensurability is an equivalence relation on the subgroups of G.
3) a) Let G be a finite group and let H ≤ G. Given g ∈ G, does gHg−1 ⊂ H imply g −1Hg ⊂ H? b) Let G be an infinite group and let H ≤ G. Given g ∈ G, does gHg−1 ⊂ H imply g −1Hg ⊂ H? Hint: consider G = GL2(Q) and let H = ("1 n 0 1# where n ∈ Z )
4) a) Let H, N be normal subgroups of a finite group G. Assume that the orders of H, N are relatively prime. Prove that xy = yx for all x ∈ H and y ∈ N, and that H ×N ∼= HN. b) Let H1,...,Hr be normal subgroups of G such that the order of Hi is relatively prime to the order of Hj for i 6= j. Prove that H1 ×...×Hr ∼= H1 ···Hr
5) Let p be a prime and let G be of order p n . Such a group is called a p-group, and it is known that for any nontrivial p-group G, the center Z(G) 6= 1. Show that G has a chain of subgroups G = G0 > G1 > G2 > ··· > Gn = 1
such that Gi is normal in G and [G : Gi ] = p i for all i. What are the composition factors of G? Hint: Use the fact that Z(G) 6= 1 to produce an element x ∈ Z(G) of order p. Prove by induction, considering the quotient group G/hxi.
6) The dihedral group D8 containing 8 elements has seven different composition series. Find all of them.
7) a) Show that an abelian group has a composition series if and only if it is finite.
b) Let F be a field and let GLn(F) denote the group of n × n invertible matrices with entries in F (the group operation is matrix multiplication). Show that GLn(F) has a composition series if and only if F is finite