Let $G$ be a group. Prove or disprove that $H=\{g^2~ |~g\in G\}$ is a subgroup of $G$
103216
The set $\{g^2 | g \in G\}$ in a group $G$
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
151727
If $|G/H|=4$ then $G$ is union of three proper subgroups
94507
Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
102622
K is a subgroup of G. Show that K is not a normal subgroup
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
158490
Prove that $G^{op} \cong G$ when $G^{op}$ is the opposite group 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 $G$.
65744
How can I Prove that H=G or K=G
146952
When can I say that $\cup H_i$ is a group?
141043
$H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
51145
Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
20679
Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
151130
When is the product of $n$ subgroups a subgroup?
56650
Prove that $(G, \circ)$ is a group if $a\circ x = b$ and $x\circ a = b$ have unique solutions
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$