Proving equality of index when $G=AB$
1251
If $[G:K]$ is finite, then $[H:H \cap K] = [G:K]$ iff $G = HK$ (Hungerford Proposition 4.8, Proof)
123555
Proving that if $ab=e$ then $ba=e$
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
17510
Proving a set is an abelian group.
83273
Show that $o(gHg^{-1})=o(H)$
103147
$\operatorname{gcd}(ab,a+b)=1$ if $a$ and $b$ are relatively prime
118030
Two subgroups of the same index must be conjugate under these assumptions
26570
Prove $\operatorname{rank}(AB) = \operatorname{rank}(B)$
5089
An element of a group has the same order as its inverse
152104
How do I prove that $ij = k?$
74452
On a uniqueness proof for solutions of $\ a + x = b\,$ in a group
83452
Prove that any conjugate of $a$ has the same order as $a$.
34973
Is it true that the order of $ab$ is always equal to the order of $ba$?
32253
How to prove $b=c$ if $ab=ac$ (cancellation law in groups)?
10483
Prove Sylvester rank inequality: $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n$
83938
If$(ab)^n=a^nb^n$ & $(|G|, n(n-1))=1$ then $G$ is abelian
33593
An example of a non-abelian group $G$ where, for all $a,b\in G$, the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$