(January 2011) Suppose that \(H\subseteq G\) is a subgroup with the property that for every \(x,y\in G\), we have \(xyx^{-1}y^{-1}\in H\). Prove that \(H\) is a normal subgroup of \(G\).
Solution
Proof. Assume g∈G and h∈H. By the given property of H, we have ghg−1h−1∈H. Since H is closed under multiplication, ghg−1 =ghg−1(h−1h) = (ghg−1h−1)h∈H as desired. QED