# Simple-complete subgroup property

From Groupprops

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

## Contents

## Definition

### Symbol-free definition

A subgroup property is said to be **simple-complete** if it satisfies the following:

- is trim, viz in every group, the trivial subgroup and the whole group satisfy property
- Every group can be embedded in a group that is -simple (that is, that satisfies the group property obtained by applying the simple group operator to ). In other words, every group can be embedded into a group that has no proper nontrivial subgroup satisfying

## Relation with other properties

### Related properties

- Finite-simple-complete subgroup property: This is a trim subgroup property where every finite group can be embedded in a finite group which is simple with respect to that property