Answer to Question #138330 in Abstract Algebra for J

Question

Assume that (G,*) is a group and that (H,*) and (K,*) are subgroups of (G,*).Prove that (H intersects K , *) is a subgroup of (G,*)

Answer 1

Let $a,b\in H\cap K$. Then $a,b\in H$ and $a,b\in K$. Since $(H,*)$ and $(K,*)$ are subgroups of

$(G,*)$, $a*b\in H$ and $a*b\in K$. By analogy for the inverse element $a^{-1}$ of the element $a$ in group $G$ we have that $a^{-1}\in H$ and $a^{-1}\in K$. Therefore, $a*b, a^{-1}\in H\cap K.$ So, we conclude that $(H\cap K, *)$ is a subgroup of $(G,*).$