Answer to Question #145960 in Abstract Algebra for J

Question

Assume that (G,*) is a group, and that H={a is an element of G: a=a^-1 }. H is a subgroup of G. Prove that H is a normal subgroup of G

Answer 1

Let $H=\{a\in G:\ a=a^{-1}\}$ be a subgroup of a group $(G,*)$. To prove that $H$ is a normal subgroup of $(G,*)$, it is sufficient to prove that $g*a*g^{-1}\in H$ for all $a\in H, g\in G.$ Let $a\in H, g\in G$ be arbitrary. Then $a^{-1}=a$, and consequently, $(g*a*g^{-1})^{-1}=(g^{-1})^{-1}*a^{-1}*g^{-1}=g*a*g^{-1}$. We conclude that $(g*a*g^{-1})^{-1}=g*a*g^{-1}$, and therefore, $g*a*g^{-1}\in H$. Thus $H$ is a normal subgroup of $(G,*)$.