Normal subgroup

A subgroup invariant under conjugation by every element of its group. Normality makes the cosets form a quotient group.

Subgroup: A subset of a group that is itself a group under the same operation. Normal subgroups are distinguished from ordinary subgroups by an additional invariance condition.
Conjugation action: The action of a group on itself by conjugation, sending x to gxg⁻¹. A normal subgroup is precisely a subset preserved by this group action.
Simple group: A nontrivial group whose only normal subgroups are the identity subgroup and itself. Normal subgroups determine whether a group is simple.
Characteristic subgroup: A subgroup preserved by every automorphism of its containing group. Characteristic subgroups are stronger than normal subgroups and are always normal.
Conjugation: The operation sending an element x to gxg⁻¹ for a fixed group element g. Normality requires the subgroup to remain unchanged under every such operation.
Quotient group: A group whose elements are cosets of a normal subgroup, with multiplication induced from the original group. Normality is the condition that makes coset multiplication well-defined.
Composition series: A finite chain of subgroups whose successive quotients are simple groups. Each step uses a normal subgroup of the preceding group.
Subnormal subgroup: A subgroup connected to its containing group by a finite chain of successive normal inclusions. Subnormality weakens normality by requiring it only at intermediate stages.
Coset: A translate of a subgroup within a group, formed by multiplying all its elements by a fixed element. Normality ensures left and right cosets coincide, enabling quotient multiplication.
First isomorphism theorem: A theorem identifying the quotient by a homomorphism's kernel with its image. It connects normal subgroups arising as kernels to quotient groups.

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