# Complemented central factor not implies direct factor

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., complemented central factor) neednotsatisfy the second subgroup property (i.e., direct factor)

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## Contents

## Statement

### Statement with symbols

It is possible to have a group and a subgroup of , such that is a complemented central factor of (i.e., and there is a subgroup of such that and is trivial) but is not a direct factor of .

## Proof

### Example of the central product of the dihedral group of order eight and the cyclic group of order four

`Further information: Central product of D8 and Z4`

Consider the group obtained as the central product of dihedral group:D8 and cyclic group:Z4, sharing a common subgroup of order two. The presentation is:

.

Let be the central factor that is the dihedral group of order eight.

Then we have:

- is a central factor of : This is by construction; is the central product of and the four-element subgroup .
- is complemented in : The subgroup is a subgroup of order two that is a complement to in .
- is not a direct factor of : There is no element of order two in outside that commutes with every element of . To see this, note that any element of outside is of the form . Its square is , since commutes with every element of . Thus, the only way the element can have order two is if , forcing or . However, in neither of these cases does centralize .