$G$, group. $a,b\in G$. Show that $|a|=|a^{-1}|; |ab|=|ba|$, and $|a|=|cac^{-1}|, \forall c\in G$.
38083
Order of conjugate of an element given the order of its conjugate
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
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 $G$.
45760
if G is generated by {a,b} and ab=ba, then prove G is Abelian
107716
Show that $G$ is cyclic
1364
Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
154001
$(g\circ f)^{-1}=f^{-1}\circ g^{-1}$
35945
Proof that if $\gcd(a,b) = 1$ and $a\mid n$ and $b\mid n$, $ab \mid n$
101543
Let $A$ be a group, where $a^2=1$, a belongs to $A$. Prove that this group is commutative.
112969
Show that $ab + bc + ac \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
139111
Prove that $a<c<b$, $\frac{f(b)-f(a)-(b-a)f'(a)}{g(b)-g(a)-(b-a)g'(a)}=\frac{f"(c)}{g"(c)}$
18528
Show that $a \star b=a \cdot b+a+b$ is binary operation for the group $\Bbb{ Q} \setminus \{-1\}$
135779
Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
59128
If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.
70394
Prove that $a * b$ is a group (Hungerford, 2nd ed., sec $7.2$, Exercise $22$)
41977
If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$