Definition The commutator [g,h]=g−1h−1gh has the following properties [h,g]=[g,h]−1, [g,h]=1 iff gh=hg.
Proposition If [x,y]∈G then xG and yG commute.
proof: xGyG=xyG by the definition of coset groups. yxG=yx(x−1y−1xy)G=xyG because G=x−1y−1xyG since x−1y−1xy∈G.
Lemma If [g,z]∈G then z−1gzG=gG=G then gz∈G.
Definition If H,K≤G their commutator [H,K]=[K,H]=⟨[h,k]∈G|h∈H,g∈G⟩ (Note! This is the group generated by commutators, not just the set of all commutators).
Definition The commutator subgroup or derived group of G is [G,G].
Proposition [G,G]charG
proof: Given α∈Aut(G), [g,h]α=[gα,hα].
Proposition (The commutator subgroup is the smallest normal subgroup N such that G/N is abelian) Given N⊴G then G/N is abelian iff [G,G]≤N.
proof: (for all g,h in G) NgNh=NhNg iff Ngh=Nhg iff [g−1,h−1]∈N.
Theorem if A≤B are both subgroups of G then [A,G]≤[B,G].
proof: We show the stronger result that the generators of [A,G] are contained in the generators of [B,G] and this is just immediate from A being a subset of B.
Theorem A factor Gi/Gi−1 of a series for G is central iff [G,Gi]≤Gi−1
proof: todo
Theorem Any index 2 subgroup is normal.
proof: Let H have index 2 in G, then there are exactly 2 cosets of it. For a∈H we aH=H=Ha so the other coset must be all the other elements so for b∈H, bH=Hb.
Theorem Any index 2 subgroup contains the commutator subgroup.
proof: If H has index 2 then G/H≃C2 is abelian so H contains G′.
Corollary The only index 2 subgroup of Sn is An.
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