Show that $Z(G) = \{ x \in G : gx = xg$ for all $g \in G$} is a subgroup of $G$.
37971
Prove that the center of a group is a normal subgroup
70373
Intersection of normal subgroups proof
2761
If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$.
46863
Show that $g$ is injective
154398
Show that H is a subgroup of G
11003
Proving that $G/N$ is an abelian group
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
23394
If $G/Z(G)$ is cyclic, then $G$ is abelian
94239
Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
156941
Subgroups of $\mathbb{Z}$
94507
Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
3756
If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
5297
If G is a finite abelian group and $a_1,...,a_n$ are all its elements, show that $x=a_1a_2a_3...a_n$must satisfy $x^2=e$.
107935
If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
138297
Let $G = GL_2\mathbb (R)$. Show that $T$ is a subgroup of $G$
152071
Group $G$ is cyclic $\iff$ every subgroup of $G$ has the form $G^k$, where $G^k=\{g^k\mid g\in G\}$
43759
Prove that the set $\mathrm{Aut}(G)$ of all automorphisms of the group $G$ with the operation of taking the composition is a group