# 1-closed subset

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This article defines a property of subsets of groups

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## Contents

## Definition

### Symbol-free definition

A nonempty subset of a group is termed a **1-closed subset** if it satisfies the following equivalent conditions:

- It is a union of subgroups
- It is a power-closed symmetric subset
- For any element in it, the cyclic subgroup generated by that element is also in it. In particular, it is a union of cyclic subgroups.