Let $G$ be a group and $g \in G$. Show that $o(g)=o(xgx^{-1})$, where $o(x)$ is the order.
83452
Prove that any conjugate of $a$ has the same order as $a$.
5069
$GL_n(\mathbb F_q)$ has an element of order $q^n-1$
56607
gcd and order of elements of group
122970
A question on $p$-groups, and order of its commutator subgroup.
127603
If $G$ be a group of order $8$ and $x$ be a element of order $4$. Prove that $x^2 \in Z(G)$
134403
Let $G$ be a group, and let $n\in\mathbb Z$. Show that $(ab)^n=a^nb^n\iff(ab)^{1-n}=a^{1-n}b^{1-n}$.
82935
Prove that f∈Θ(g)if and only ifO(f) =O(g).
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
101602
consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n
83654
Order of an element in a finite cyclic group
55976
Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
82037
Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$?
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
158490
Prove that $G^{op} \cong G$ when $G^{op}$ is the opposite group of G