= Classification of finite simple groups
The <classification of finite simple groups> states that every <finite simple group> is isomorphic to a cyclic group of prime order, an <alternating group> $A_n$ with $n\ge5$, a finite simple <group of Lie type>, or one of twenty-six <sporadic simple groups>. In the Lie-type families one takes the appropriate simple groups, respecting the exceptional small parameters and quotient constructions.<Group Theory (v4.01) by James S. Milne>[section 3, The Hölder program, part A, printed pages 52–53]{ref}
For example, $C_7$ is a simple <abelian group>, while $A_5$ is a simple nonabelian group of order sixty. The <symmetric group> $S_3$ is not simple: its subgroup $A_3$ is a proper nontrivial <normal subgroup>. The theorem lists the simple building blocks that occur as <composition factors>. It does not classify all <finite groups>: different <group extensions> can assemble the same factors into nonisomorphic groups.
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