For every element $g$ in a group $G$ of order $23$, there is $h\in G$ such that $g=h^2$.
2771
A group has odd number of elements iff each element is a square, $a=b^2$.
116080
Let G be a finite group with more than one element. Show that G has an element of prime order
14379
Let $H \leq G$. If $G = H H^x$ then $G=H$.
103216
The set $\{g^2 | g \in G\}$ in a group $G$
129452
Groups $G,H$ with surjective homomorphisms that are quotient of each other
20679
Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
26398
Group of even order contains an element of order 2
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
106277
Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
114590
Prove that any finite group $G$ of even order contains an element of order $2.$
89659
$H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute
151025
If G is a group of order n=35, then it is cyclic
2731
Groups of even order have an element whose order is 2.
67554
In a group, the number of non-self-inverse elements is even
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
107935
If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
52121
Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.