Answer to Question #297119 in Abstract Algebra for Luqman

Question

If H and K is subgroup prove H intersection K is subgrop.

Answer 1

Let $H$ and $K$ are subgroups. Let us prove that $H \cap K$ is subgrop.

Let $a,b\in H\cap K.$ Then $a,b\in H$ and $a,b\in K.$ Since $H$ and $K$ are subgroups, $ab,a^{-1}\in H$ and $ab,a^{-1}\in K.$ Therefore, $ab,a^{-1}\in H\cap K,$ and we conclude that $H \cap K$ is subgrop.