$ [x,y]=y, [y,z]=z, [z,x]=x$ implies $x=y=z=e$ in a group
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
132127
Proving a direct product of groups is a group
63359
Show that the product of all group elements is $e$ or $u$ if $G$ has exactly one involution $u$
125559
In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian
142631
Show that A has the same cardinality as $A\setminus\{x,y,z\}$
56413
$f(g(x))=g(f(x))$ implies $f(c)=g(c)$ for some $c$
114522
Prove that identity is unique in a group
15633
If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
20571
In a monoid, does $x \cdot y=e$ imply $y \cdot x=e$?
7316
How to prove that $z\gcd(a,b)=\gcd(za,zb)$
56245
Let H be a subgroup of a group G. Prove that the following statements are equivalent.
37414
Question on Groups $G=\langle x,y|x^4=y^4=e,xyxy^{-1}=e\rangle$
152604
For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?
5089
An element of a group has the same order as its inverse
157915
Isomorphism between the group $(\mathbb Z[x], +)$ and $(\mathbb Q_{>0}, .)$
3203
Quotient group $\mathbb Z^n/\ \text{im}(A)$
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 $x^2=e$.