# Complete solvable group

From Groupprops

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: complete group and solvable group

View other group property conjunctions OR view all group properties

## Definition

A **complete solvable group** is a group that satisfies *both* these conditions:

- It is a complete group, i.e., it is both a centerless group and a group in which every automorphism is inner.
- It is a solvable group.

## Relation with other properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

solvable group in which every automorphism is inner | we no longer require the group to be centerless | there exist nilpotent nontrivial groups, such as cyclic group:Z2, in which every automorphism is inner | |FULL LIST, MORE INFO | |

complete group | there are complete non-solvable groups such as symmetric group:S5 | complete group|complete | ||

solvable group | Group whose automorphism group is solvable, Solvable group in which every automorphism is inner|FULL LIST, MORE INFO | |||

group in which every automorphism is inner | cyclic group:Z2 | Solvable group in which every automorphism is inner|FULL LIST, MORE INFO | ||

group whose automorphism group is solvable | |FULL LIST, MORE INFO |