# Quotient-powering-invariance is union-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., quotient-powering-invariant subgroup) satisfying a subgroup metaproperty (i.e., union-closed subgroup property)

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## Contents

## Statement

Suppose is a group and is a collection of quotient-powering-invariant subgroups of . Suppose the union is a subgroup of (and hence is also the same as the join of subgroups . Then, this union of also a quotient-powering-invariant subgroup of .

## Related facts

## Facts used

## Proof

Note that existence of roots is guaranteed (see divisibility is inherited by quotient groups). The part we need to establish is uniqueness. **Note also that by Fact (1), the union must be a normal subgroup if it is a subgroup,** hence we will assume this in our setup.

**Given**: is a group and is a collection of quotient-powering-invariant subgroups of . The union is a subgroup of . is powered over a prime . Two elements that are in the same coset of (concretely, ). are respectively the unique solutions to and .

**To prove**: and are in the same coset of .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | There exists such that . In other words, there exists such that and are in the same coset of . | is the union of , . | Follows directly from given. | ||

2 | and are in the same coset of . | is -powered, is quotient-powering-invariant in . | Step (1) | Direct from given and Step (1). Note that quotient-powering-invariant in a -powered group means that the roots of elements in the same coset are in the same coset. | |

3 | and are in the same coset of . | is the union of . | Step (2) | Since , being in the same coset of implies being in the same coset of . |