# Cyclic isomorph-containing subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: isomorph-containing subgroup with a group property (itself viewed as a subgroup property): cyclic group

View a complete list of such conjunctions

## Contents

## Definition

A subgroup of a group is termed a **cyclic isomorph-containing subgroup** if is a cyclic group and is an isomorph-containing subgroup of , i.e., every subgroup of isomorphic to is contained in .

## Relation with other properties

### Stronger properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Cyclic homomorph-containing subgroup |

### Weaker properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

1-automorphism-invariant subgroup | | | |||

* Cyclic 1-automorphism-invariant subgroup | | | |||

Quasiautomorphism-invariant subgroup | | | |||

* Cyclic quasiautomorphism-invariant subgroup | | | |||

Cyclic characteristic subgroup | | | |||

Cyclic normal subgroup | | | |||

Characteristic subgroup | | | |||

Normal subgroup | | |