The First Isomorphism Theorem

The First Isomorphism Theorem

Letandbe groups, and letbe a homomorphism. Then:

1. The kernel ofis a normal subgroup of

2. The image ofis a subgroup ofand

3. The image ofis isomorphic to the quotient groupwhose elements areThe identity inis

In particular, ifis surjective thenis isomorphic to

In fact given any normal subgroupwe can define a homomorphismsuch that is surjective (onto) by construction, and well defined since

Example:

Letis a normal subgroup ofand

The kernel ofis the set of elements of g that are sent byto the identity

Labellingbywe can writeand