Coprime implies one is solvable: Difference between revisions
(New page: ==Statement== If two finite groups have relatively prime orders, then one of the groups is solvable. ==Proof== This follows directly from the [[Feit-Thompson ...) |
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If two [[finite group]]s have relatively prime [[order]]s, then one of the groups is [[solvable group|solvable]]. | If two [[finite group]]s have relatively prime [[order]]s, then one of the groups is [[solvable group|solvable]]. | ||
==Related facts== | |||
===Applications=== | |||
* [[Hall retract implies order-conjugate]] (part of the [[Schur-Zassenhaus theorem]]) | |||
* [[Sylow's theorem with operators]] | |||
For a complete list of applications, see [[:Category:Facts about groups of coprime order whose proof requires the assumption that one of them is solvable]]. | |||
==Proof== | ==Proof== | ||
This follows directly from the [[Feit-Thompson theorem]]. | This follows directly from the [[Feit-Thompson theorem]]. | ||
Latest revision as of 13:53, 20 August 2011
Statement
If two finite groups have relatively prime orders, then one of the groups is solvable.
Related facts
Applications
- Hall retract implies order-conjugate (part of the Schur-Zassenhaus theorem)
- Sylow's theorem with operators
For a complete list of applications, see Category:Facts about groups of coprime order whose proof requires the assumption that one of them is solvable.
Proof
This follows directly from the Feit-Thompson theorem.