(January 2012) Let \(G\) be a group, and let \(H,K\subseteq G\) be normal subgroups of \(G\). Prove that \(H\cap K\) is a normal subgroup of \(G\). You may assume without proof that \(H\cap K\) is a subgroup.
Solution
Proof. Givena∈H∩K,g∈G, we know thatgag−1∈H andgag−1∈K sinceH andK are normal. Thus,gag−1∈H∩K, so H∩K is normal in G as desired. QED