Prove the if $o(g) =x$ and $o(h) = y$ then $o(gh) = xy$
25713
If $a$ and $b$ commute and $\text{gcd}\left(\text{ord}(a),\text{ord}(b)\right)=1$, then $\text{ord}(ab)=\text{ord}(a)\text{ord}(b)$.
142375
Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
40609
Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
93596
If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$
62455
Prove Equivalence Relation in G
151047
a non trivial example: $G/H$ is isomorphic to $G$
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
81671
Proof of identity $\sqrt {xy} = \sqrt x \sqrt y$ for $x,y \in \mathbb R^+$
56245
Let H be a subgroup of a group G. Prove that the following statements are equivalent.
15089
Big-O: If $f(n)=O(g(n))$, prove $2^{f(n)}=O(2^{(g(n)})$
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
63535
Prove that if $A – C = B – C$ and $A\cap C = B\cap C$ then $A = B$
89659
$H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
55016
Prove that if ($d(x,y)=0$ iff $x=y$) and if $d(x,z)\leq d(x,y)+d(z,y)$ then $d$ is a metric
76694
Proving that $UT=TU$ iff $U=g(T)$ for some polynomial $g$