Finite simple group

A finite group with more than one element whose only normal subgroups are the identity subgroup and the group itself. It has no nontrivial proper quotient groups.

Group (mathematics): A set with an associative operation, an identity element, and an inverse for every element. Finite simple groups are groups whose normal-subgroup structure admits no proper nontrivial part.
Classification of finite simple groups: The theorem that lists all finite simple groups through several infinite families and 26 sporadic groups. It identifies every possible finite simple group, not merely common examples.
Évariste Galois: A French mathematician whose work established foundational connections between groups and polynomial equations. Galois theory helped make group structure a central object of algebra.
Composition series: A chain of subgroups whose successive quotients are simple groups. Every finite group can be decomposed into finite simple composition factors.
Abelian group: A group whose operation is commutative. Among finite simple groups, only cyclic groups of prime order are abelian.
Normal subgroup: A subgroup invariant under conjugation by every element of its group. Normality is the exact condition used to define simplicity.
Cyclic group of prime order: A cyclic group whose order is a prime number. These are exactly the abelian finite simple groups.
Camille Jordan: A French mathematician who developed group theory and introduced the Jordan–Hölder theorem. His work formalized composition series, whose factors are finite simple groups.
Jordan–Hölder theorem: A theorem stating that a group's composition factors are unique up to order and isomorphism. It makes the simple factors intrinsic to the finite group being analyzed.
Simple group: A nontrivial group with no normal subgroups other than the identity subgroup and itself. The broader notion includes infinite groups as well as finite ones.

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