Prove that every element of a group G can be represented as $g = x^{-1}(xT)$ for some x $\in$ G?
103801
$xT=x$ iff $x=e$ implies that every $g$ may be written as $x^{-1}(xT)$ for some $x\in G$.
112974
The $h-$th power of every element in a finite group of order $h$ is the identity element of the group
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 $G$
13807
Is there a way to prove that the order of an element in a Group divides the order of the Group, WITHOUT USING LAGRANGE'S
92057
Prove that for every odd integer n, there is a group...
14097
A semigroup $X$ is a group iff for every $g\in X$, $\exists! x\in X$ such that $gxg = g$
63359
Show that the product of all group elements is $e$ or $u$ if $G$ has exactly one involution $u$
31252
"Efficient version" of Cayley's Theorem in Group Theory
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
152604
For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?
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 $x^2=e$.
76694
Proving that $UT=TU$ iff $U=g(T)$ for some polynomial $g$
71420
Prove that a group is abelian.
24181
Showing that $G$ is a group under an alternative operation.
94239
Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
23517
Presentation $\langle x,y,z\mid xyx^{-1}y^{-2},yzy^{-1}z^{-2},zxz^{-1}x^{-2}\rangle$ of group equal to trivial group
135465
Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$