A finite quotient is the image of a group under a surjective group homomorphism to a finite group. A group has no nontrivial finite quotients exactly when every homomorphism from it to a finite group is trivial. This property does not prohibit finite or finite-quotient-bearing subgroups.
An abelian quotient is a quotient group that is an abelian group. It exists only for a normal subgroup , and it is abelian exactly when contains the commutator subgroup. Thus every abelian quotient factors through the abelianization .
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In group theory, a **quotient group** (or factor group) is a way of constructing a new group from an existing group by partitioning it into disjoint subsets, called cosets, that are determined by a normal subgroup. Here's how it works, step by step: 1. **Group**: Let \( G \) be a group, which is a set equipped with a binary operation satisfying the group axioms (closure, associativity, identity element, and inverses).
Ultimate explanation: math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426#3732426
Does not have to be isomorphic to a subgroup:
This is one of the reasons why the analogy between simple groups of finite groups and prime numbers is limited.