The First Isomorphism Theorem
The First Isomorphism Theorem
Letandbe groups, and letbe a homomorphism. Then:
The kernel ofis a normal subgroup of
The image ofis a subgroup ofand
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