# Powering-injectivity is central extension-closed

This article gives the statement, and possibly proof, of a group property (i.e., powering-injective group for a set of primes) satisfying a group metaproperty (i.e., central extension-closed group property)

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## Statement

Suppose is a group and is a central subgroup of . Suppose is a prime number such that:

- is -powering-injective, i.e., is injective from to itself.
- The quotient group is -powering-injective, i.e., is injective from to itself.

Then, the whole group is -powering-injective.

## Related facts

## Proof

**Given**: A group , a prime number . A central subgroup of such that in both and viewed separately, the map is injective. Two elements with .

**To prove**: .

**Proof**: Let be the quotient map.

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

1 | . | , and the map is injective in . | We have , hence . Thus, , so are elements of with the same power. Thus, they must be equal. | ||

2 | The element is an element of . | Step (1) | is the identity element of , so . | ||

3 | is the identity element of . | , is central in . | Step (2) | We have . Thus, . By the centrality of , we get math>a^p = u^pb^p</math>. We also have . This gives , so is the identity element of , and hence also of the subgroup . | |

4 | is the identity element of . | is injective in . | Step (3) | Step-given direct. | |

5 | . | Steps (2), (4) | Step-combination direct. |