Abstract Algebra: Theory and Applications · Problem 15.4.26

Rings — Thomas Judson

Mathematics — Abstract Algebra · sylow-theorems

Problem

Let GG be a group. Prove that G′=⟨aba−1b−1:a,b∈G⟩G' = \langle aba^{-1}b^{-1} : a, b \in G \rangle is a normal subgroup of GG and G/G′G/G' is abelian. Find an example to show that {aba−1b−1:a,b∈G}\{aba^{-1}b^{-1} : a, b \in G\} is not necessarily a group. Hint. Let aG′,bG′∈G/G′aG', bG' \in G/G'. Then (aG′)(bG′)=abG′=ab(b−1a−1ba)G′=(abb−1a−1)baG′=baG′(aG')(bG') = abG' = ab(b^{-1}a^{-1}ba)G' = (abb^{-1}a^{-1})baG' = baG'.

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