Abelian Normal Subgroup, Intersection, and Product of Groups
Let $G$ be a group and let $A$ be an abelian subgroup of $G$ with $A \triangleleft G$.
(That is, $A$ is a normal subgroup of $G$.)
If $B$ is any subgroup of $G$, then show that
\[A \cap B \triangleleft AB.\]
First of all, since $A \triangleleft G$, the […]
Commutator Subgroup and Abelian Quotient Group
Let $G$ be a group and let $D(G)=[G,G]$ be the commutator subgroup of $G$.
Let $N$ be a subgroup of $G$.
Prove that the subgroup $N$ is normal in $G$ and $G/N$ is an abelian group if and only if $N \supset D(G)$.
Recall that for any $a, b \in G$, the […]
Any Subgroup of Index 2 in a Finite Group is Normal
Show that any subgroup of index $2$ in a group is a normal subgroup.
Left (right) cosets partition the group into disjoint sets.
Consider both left and right cosets.
Let $H$ be a subgroup of index $2$ in a group $G$.
Let $e \in G$ be the identity […]