Let $G$ be a group where $(ab)^3=a^3b^3$ . Prove that $\{ x^3 \mid x \in G \}$ is a normal subgroup of $G$.
111886
Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.
80601
Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
61732
Prove any group with 3 elements is isomorphic to $\mathbb{Z}_3$
99630
For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
97928
Normal subgroup of a normal subgroup
134403
Let $G$ be a group, and let $n\in\mathbb Z$. Show that $(ab)^n=a^nb^n\iff(ab)^{1-n}=a^{1-n}b^{1-n}$.
37971
Prove that the center of a group is a normal subgroup
12589
Show that every subgroup of the quaternion group Q is a normal subgroup of Q
93596
If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
32735
If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$.
123610
How to show that the commutator subgroup is a normal subgroup
37379
Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$. Show that if $G$ is generated by $M\cup N$ then $G\cong M \times N$.
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
42891
Prove that $Z(G)$ which is the center of $G$ is a subgroup of $G$
65926
Group of order $1575$ having a normal sylow $3$ subgroup is abelian.
102622
K is a subgroup of G. Show that K is not a normal subgroup