# Center not is divisibility-closed

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) doesnotalways satisfy a particular subgroup property (i.e., divisibility-closed subgroup)

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Contents

## Statement

It is possible to have a group such that there exists a natural number such that:

- is -divisible: For every , there exists satisfying .
- The center is not -divisible: There exists an element such that there is no satisfying .

## Related facts

### Opposite facts

## Proof

### Example of a Lie group

We will construct an example for the case .

Let be the group of unit quaternions under multiplication, i.e., it is the group . Explicitly:

with multiplication given by multiplication of quaternions.

We claim the following:

#### Every element of the group is a square

- Case : In this case, the element is a square root.
- All other cases: The following element works as a square root:

Note that the intuition is as follows: it turns out that any element with is a square root of the element . Thus, every element of the group can be put inside a copy of inside where the imaginary part is the normalized imaginary part of the element. Now, taking the square root is like taking square roots inside .

Note also that with the exception of -1, every element has only two square roots, which are negatives of each other.

#### There is an element of the center that is not the square of any element in the center

The center is the subgroup . The element in the center does not have any square root in the center. As noted above, its set of square roots is the set:

None of these elements is central.