Let $A$ and $B$ be normal subgroups of a group $G$. Suppose $A\cap B=\{e\}$, where $e$ is the unit element of the group $G$.
Show that for any $a \in A$ and $b \in B$ we have $ab=ba$.
An isomorphism from a group $G$ to itself is calledan automorphismof $G$.
The set of all automorphism is denoted by $\Aut(G)$.
Definition (characteristic subgroup).
A subgroup $H$ of a group $G$ is called characteristic in $G$ if for any $\phi \in \Aut(G)$, we have $\phi(H)=H$. In words, this means that each automorphism of $G$ maps $H$ to itself.
Prove the followings.
(a) If $H$ is characteristic in $G$, then $H$ is a normal subgroup of $G$.
(b) If $H$ is the unique subgroup of $G$ of a given order, then $H$ is characteristic in $G$.
(c) Suppose that a subgroup $K$ is characteristic in a group $H$ and $H$ is a normal subgroup of $G$. Then $K$ is a normal subgroup in $G$.