(January 2016) Let \(G\) be a group, let \(H\subseteq G\) be a subgroup, and define the set \[ K=\{x\in G\mid Hx=xH\}. \] Prove that \(K\) is a subgroup of \(G\).
Solution
Proof. We prove thatK is a subgroup of G as follows. (1) Let e be the identity element of G. Then He =H =eH, so e∈K. (2) Given a,b∈K, we have H(ab) = (Ha)b = (aH)b =a(Hb) =a(bH) = (ab)H. So ab∈K. (3) Given a∈K, we have Ha =aH. Multiplying by a−1 on the left and the right gives a−1(Ha)a−1 =a−1(aH)a−1 =⇒ a−1H(aa−1) = (a−1a)Ha−1, which implies a−1H =Ha−1. Thus a−1∈K, proving that K is a subgroup. QED