Category:Facts about groups of coprime order whose proof requires the assumption that one of them is solvable
This category lists facts whose statement involves a pair of groups of coprime order. The proof requires the assumption that at least one of the groups is solvable. Such facts are true because given two groups of coprime order, one of them is solvable, but the only known proof of this is using the odd-order theorem: any group of odd order is solvable. The odd-order theorem isn't considered elementary because its proof is rather involved, and requires a mix of techniques including character theory.
Pages in category "Facts about groups of coprime order whose proof requires the assumption that one of them is solvable"
The following 6 pages are in this category, out of 6 total. The count includes redirect pages that have been included in the category. Redirect pages are shown in italics.
- Centralizer of coprime automorphism group in homomorphic image equals image of centralizer if either is solvable
- Centralizer-commutator product decomposition for finite groups
- Commutator of finite group with coprime automorphism group equals second commutator
- Hall retract implies order-conjugate
- Schur-Zassenhaus theorem
- Sylow's theorem with operators