# Category:Facts about groups of coprime order whose proof requires the assumption that one of them is solvable

From Groupprops

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.

### V

- 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