How to use automorphism of Group to define automorphism of its factor group?

Question

Let $G$ be a group and $T$ an automorphism of $G.$ If $N$ is a normal subgroup of $G$ such that $T(N)\subset N,$ then how to use this $T$ to define an automorphism of $G/N?$ In this problem I get stuck. I have no idea to how to start a solution. Any help will be appreciated.

Answer 1

Here is a natural roadmap:

Let $\overline G=G/N$ and define $\overline T: \overline G \to \overline G$ by $\overline T(\overline g) = \overline {T(g)}$. Now prove:

• $\overline T$ is well-defined.

• $\overline T$ is a homomorphism.

• $\overline T$ is an automorphism.

The only real nontrivial point is that $\overline T$ is injective. For that, we need $T(N)=N$.