# Finite group with cyclic quotient of automorphism group by class-preserving automorphism group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and group with cyclic quotient of automorphism group by class-preserving automorphism group
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## Definition

A finite group with cyclic quotient of automorphism group by class-preserving automorphism group is a finite group that is also a group with cyclic quotient of automorphism group by class-preserving automorphism group, i.e., a group such that the quotient group of its automorphism group by the normal subgroup comprising the class-preserving automorphisms is a cyclic group (and in particular, a finite cyclic group).

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finite complete group
Finite group whose outer automorphism group is cyclic

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finite group whose automorphism group has equivalent actions on the sets of conjugacy classes and irreducible representations
Finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group