Odd-order implies solvable

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., odd-order group) must also satisfy the second group property (i.e., solvable group)
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Template:Odd-order-only finite-group statement This fact is useful in work leading up to the Classification of finite simple groups

History

This result was proved by Feit and Thompson, and is called the Feit-Thompson Theorem or the Odd order theorem.

Statement

Verbal statement

Any finite group of odd order is solvable. Equivalently, any finite simple non-Abelian group has even order.

Property-theoretic statement

The property of being an odd-order group is a stronger property than the property of being solvable.

Applications

• Coprime implies one is solvable: If two finite groups have relatively prime orders, then one of them is solvable.

Proof

The proof of the odd-order theorem is nontrivial and cannot be put into the wiki page.