Necessary and sufficient condition for a subgroup to be factor

Question

Let $G =A \times B$, where I will call $A$ and $B$ as factors of group $G$.

For an example $Z_{21} =Z_7 \times Z_3$, $Z_3$ is a factor of group $Z_{21}$.

Given a group $G$ and a subgroup $A$ how to check whether $A$ is a factor of $G$ or not.

Here is my idea, find the $G/A$ and then set an homomorphism from $G/A$ to the group $G$.

Question : What is neccessary and sufficient conidtion for a subgroup to be a factor?

Answer 1

$A$ is a direct factor of $G$ if and only if the following two conditions hold:

(i) $AC_G(A)=G$; and

(ii) $Z(A)$ has a complement in $C_G(A)$.

Note that $G = A \times B$, where $B$ is any complement of $Z(A)$ in $C_G(A)$.

Answer 2

For abelian group $G$, it's necessary and sufficient for a subgroup $A$ to be a factor that there is a homomorphism $G/A\to G$ such that when composed with the canonical projection $G\to G/A$, the homomorphism $$ G/A\to G\to G/A $$ is the identity on $G/A$. Dually, it's necessary and sufficient that there is a homomorphism $G\to A$ such that when composed with the inclusion $A\to G$, the homomorphism $$ A\to G\to A $$ is the identity on $A$.

In your example, for instance, we have $A = \Bbb Z_3$ and $G = \Bbb Z_{21}$. The relevant homomorphism $G\to A$ is given by $$ 1, 4, 7, 10, 13, 16, 19\mapsto1\\ 2, 5, 8, 11, 14, 17, 20\mapsto 2\\ 0, 3, 6, 9, 12, 15, 18\mapsto 0 $$ (also known as "reducing modulo $3$") while the relevant homomorphism $G/A\cong\Bbb Z_7\to \Bbb Z_{21}$ is given by $$ a\mapsto 3a $$