$3$ does not divide $|G| \implies \forall \ g \in G, g=h^3$ , for some $h$ in $G$
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
133627
If $\phi(g)=g^3$ is a homomorphism and $3 \nmid |G|$, $G$ is abelian.
142375
Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
51373
$G$ non abelian, order $p^3$ ($p$ prime). Suppose that the center is $p^2$, prove that $\exists\ x$ outside of the center, of order p
19202
$\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$ if and only if $d$ divides $n$
63162
Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?
86542
Group of positive rationals under multiplication not isomorphic to group of rationals
14379
Let $H \leq G$. If $G = H H^x$ then $G=H$.
65744
How can I Prove that H=G or K=G
46355
If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
151047
a non trivial example: $G/H$ is isomorphic to $G$
44307
Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$
58654
If $|A| > \frac{|G|}{2} $ then $AA = G $
4883
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
111024
Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$?