Answer to Question #139751 in Abstract Algebra for J

Question

If (H,*) is a subgroup of (G,*), and a * b is an element of H, must a is an element of H and b is an element of H? Explain your answer.

Answer 1

Let $(G,*)$ be a group containing at least 2 elements, and $e$ be the identity of $(G,*)$. Let $H=\{e\}$. Recall that $e^{-1}=e$ and $e*e=e.$ Then $(H,*)$ is a subgroup of $(G,*)$. If $a\in G$ and $a\ne e$, then $a^{-1}\ne e$ as well. Put $b=a^{-1}$. Then $a*b=a*a^{-1}=e\in H$, but $a\notin H$ and $b\notin H$.