Let $G$ be a group and let $\Phi\colon G \to G$ be an isomorphism. Define $H = \{ a \in G\ |\ \Phi(a) = a^{-1} \}$
73113
Groups and Symmetries
43759
Prove that the set $\mathrm{Aut}(G)$ of all automorphisms of the group $G$ with the operation of taking the composition is a group
29445
Trying to prove that $H=\langle a,b:a^{3}=b^{3}=(ab)^{3}=1\rangle$ is a group of infinite order.
120971
(Revised) Prove that $\phi$ is a group homomorphism and find the kernel.
65744
How can I Prove that H=G or K=G
142739
Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$
139835
Every Finite Group is Isomorphic to a Subgroup of $A_n$
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
158975
Isomorphism of centers of isomorphic groups
154398
Show that H is a subgroup of G
73988
Characterize the groups $G$ for which the map $\iota: G \to G$, sending $x \mapsto x^{-1}$ for all $x \in G$, is an automorphism of $G$
39165
Suppose $G$ is a group and $H,K \trianglelefteq G$. If $H \cap K =1$ the $hk=kh$ for all $k \in K, h \in H$?
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
45179
When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
130081
Show that $A_n$ is the kernel of a group homomorphism of $S_n \rightarrow \{−1,1\}$.
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
5089
An element of a group has the same order as its inverse