A simple group is a Group whose Normal Subgroups (Invariant Subgroups) are Order one or the whole of the original Group. Simple groups include Alternating Groups, Cyclic Groups, Lie-Type Groups (five varieties), and Sporadic Groups (26 varieties, including the Monster Group). The Classification Theorem of finite simple groups states that such groups can be classified completely into the five types:

- 1. Cyclic Groups of Prime Order,
- 2. Alternating Groups of degree at least five
- 3. Lie-Type Chevalley Groups,
- 4. Lie-Type (Twisted Chevalley Groups or the Tits Group), and
- 5. Sporadic Groups.

Burnside's Conjecture states that every non-Abelian Simple Group has Even Order.

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1999-05-26