Prove that the order of an element $g$ in $G$ is equal to the order of its inverse.
5089
An element of a group has the same order as its inverse
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
37061
The order of $ab$ when $a,b$ commute
46260
Prove that a group generated by two elements of order $2$, $x$ and $y$, is isomorphic to $D_{2n}$, where $n = |xy|.$
46863
Show that $g$ is injective
146421
How to prove a total order has a unique minimal element
88344
Order of a group and subsequent subgroups
110313
Prove that $\gcd(f(x),g(x)) = 1$.
81116
$gh = hg, \ \gcd(|g|, |h|) = 1\Rightarrow|gh| = |g||h|\ \ (|a|$ = order of element $a)$
131946
Order of an Element Under a Homomorphism
92873
To prove in a Group Left identity and left inverse implies right identity and right inverse
30003
Proving that Order for N is Anti-symmetric
15633
If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
119808
Proving that $n$-cycle has order $n$
65744
How can I Prove that H=G or K=G
2731
Groups of even order have an element whose order is 2.
150156
Proving that $S_n$ has order $n!$