Let $\phi:G\rightarrow G$ be a group homomorphism. Prove that there exists $n$ such that $G\cong ker(\phi^n)\bigoplus \phi^n(G)$
135808
Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$
7890
Prove the map $\Phi:G\to\operatorname{Aut}G$ with $g\mapsto (x\mapsto g^{-1}xg)$ is an homomorphism.
67199
Isomorphism $H_n(D^n, S^{n - 1};A)$ and $A$
99630
For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
13764
Homomorphisms from $\mathbb{Z} \rightarrow S_3$.
7967
Prove that the order of the cyclic subgroup $\langle g^k\rangle $ is $n/{\operatorname{gcd}(n,k)}$
120971
(Revised) Prove that $\phi$ is a group homomorphism and find the kernel.
126930
Finding all group homomorphisms of $S_n\to\mathbb{C}^*$ and $A_n\to\mathbb{C}^*$
90616
When is mapping $g$ to $g^{-1}$ a group homomorphism?
63162
Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?
144065
Proving $\phi(G)$ is cyclic if $\phi$ : $G \to H$ is group homomorphism and $G$ is cyclic
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
124140
Quick way to find the number of the group homomorphisms $\phi:{\bf Z}_3\to{\bf Z}_6$?
90010
Can we ascertain that there exists an epimorphism $G\rightarrow H$?
158282
Finite order of an element $f(g)$ of a homomorphism $f$
125582
Let $\phi:G_1\to G_2$ be a group homomorphism. Show $\phi(g^{-1})=(\phi(g))^{-1}$.