For elements $a,b \in G$, group that acts on the set $X$, show that $a$ and $b^{-1}ab$ have the same numbers of fixed points in $X$
133179
Fixed points under action of conjugate group elements
156850
Fix-Point Theorem Proof.
83985
Equivalent statements to fixed-point theorem
63691
G acts on X transitively, then there exists some element that does not have any fixed points
56650
Prove that $(G, \circ)$ is a group if $a\circ x = b$ and $x\circ a = b$ have unique solutions
85011
Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
80601
Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
83784
Fixed Point of $x_{n+1}=i^{x_n}$
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
11003
Proving that $G/N$ is an abelian group
92430
Show that the group is cyclic.
55976
Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?
32253
How to prove $b=c$ if $ab=ac$ (cancellation law in groups)?
152904
Groups with Same Number Of Elements and Subgroups
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
158498
$A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$
24181
Showing that $G$ is a group under an alternative operation.