Normal subgroup of finite index

Symbol-free definition
A normal subgroup of finite index in a group is a subgroup satisfying the following equivalent conditions:


 * 1) It is normal and its index in the whole group is finite
 * 2) It is the kernel of a homomorphism to a finite group
 * 3) It is the normal core of a subgroup of finite index

Equivalence of definitions
The equivalence of definitions (1) and (2) follows from the first isomorphism theorem. The equivalence with definition (3) follows from Poincare's theorem.