Let $G$ be a finite group and $A \le G$. If $G=AA^g$ for some $g$ then $G=A$.
14379
Let $H \leq G$. If $G = H H^x$ then $G=H$.
34724
For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
90437
Let $G = \langle a\rangle$ a cyclic group of order m. Prove that G is isomorphic with $Z_m$.
41977
If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$
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 $G$.
27905
Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G $ is an abelian group.
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
1364
Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
117468
$o(g^{k})=\frac{o(g)}{\gcd(o(g),k)}$ order of a power of a group element.
65613
$G$ is a finite group and $T$ an automorphism of $G$ such that $T$ sends more than $\frac{3}{4}|G|$ elements to their inverse.
19458
Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.
47179
Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
14097
A semigroup $X$ is a group iff for every $g\in X$, $\exists! x\in X$ such that $gxg = g$
150199
How to prove for a finite group that $|\langle g \rangle|=o(g)$?
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
21673
If $|G| = p^n$ then $G$ has a subgroup of order $p^m$ for all $0\le m <n.$