Simple group

A group whose only normal subgroups are the trivial subgroup and the group itself. Simple groups are the basic building blocks of finite groups through composition series.

Normal subgroup: A subgroup invariant under conjugation by every element of its group. Simplicity is defined by allowing only the trivial subgroup and the whole group to be normal.
Cyclic group of prime order: A cyclic group with a prime number of elements. Every nonidentity element generates the whole group, making it simple.
Évariste Galois: A French mathematician whose work connected polynomial equations with permutation groups. His theory made group structure central to understanding solvability by radicals.
Abelian group: A group whose operation is commutative. Among nontrivial abelian groups, simplicity occurs exactly at prime order.
Quotient group: A group formed by treating all elements of a normal subgroup as the identity. Every proper nontrivial normal subgroup would produce a nontrivial quotient.
Alternating group: The group of even permutations on a finite set. Alternating groups of degree at least five are simple.
Camille Jordan: A French mathematician who developed foundational work on permutation groups and composition series. His work helped establish the role of simple factors in finite group structure.
Solvable group: A group whose derived series eventually reaches the trivial subgroup. Nonabelian simple groups obstruct solvability, while prime-order simple groups are solvable.
Composition series: A chain of subgroups whose successive quotients are simple groups. Simple groups appear as the indivisible factors in this chain.
Projective special linear group: A group induced by determinant-one linear transformations after scalar matrices are factored out. Most groups in this family provide major examples of finite simple groups.

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