If $f$ has no non trivial fixed points and $f\circ f$ is the identity then $f(x)=x^{-1}$ and $G$ is abelian for $f$ an automorphism of $G$
158396
Finite $G$ with involutory automorphism $\alpha$ with no nontrivial fixed points. Proving properties of $\alpha$.
40321
$\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$.
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
45432
Showing that $f:(0,2\pi]\to S^1$ is not a homeomorphism
10024
How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$? (inverse of composition)
127310
Suppose $\gcd(\deg(f),\deg (g))=1$. Show that $g(x)$ is irreducible in $k(\alpha)[X]$.
13761
Automorphisms in $\mathbb{Z}_n$
58642
If $g \circ f$ is the identity function, then which of $f$ and $g$ is onto and which is one-to-one?
95299
What can you say about $f$ and $g$ in the case that $fg$ is 1) Injective, 2) Surjective - Cohn - Classic Algebra Page 15
36402
Prove that $f(x) = \cos(x)$ has a unique fixed point.
11202
If $g \circ f$ is injective, so is $g$
92493
If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$
4228
Prove that the only automorphism of $\mathbb{Q}(\sqrt[3]{2})$ is the identity automorphism.
70912
If $g\circ f$ is injective and $f$ is surjective then $g$ is injective
100134
Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC
65900
Show that a finite group with certain automorphism is abelian
40993
Let $f:G\to G'$ be a surjective homomorphism. Prove that if $G$ is cyclic, then $G'$ is cyclic.