# Exponent of semidirect product may be strictly greater than lcm of exponents

From Groupprops

## Contents

## Statement

### In terms of internal semidirect products

It is possible to have a finite group , a complemented normal subgroup of with a complement (so is an internal semidirect product of and ) such that the exponent of is strictly greater than the lcm of the exponents of and .

## Related facts

- Exponent of extension group is a multiple of lcm of exponents of normal subgroup and quotient group
- Exponent of extension group divides product of exponents of normal subgroup and quotient group
- Exponent of semidirect product may be strictly less than product of exponents

## Proof

### Example of dihedral group

`Further information: dihedral group:D8, subgroup structure of dihedral group:D8`

Consider to be the dihedral group:D8:

This has eight elements:

Suppose is one of the Klein four-subgroups of dihedral group:D8:

and take as one of the non-normal subgroups of dihedral group:D8 that is *not* contained in :

We note that:

- The exponent of is 4.
- The exponent of is 2.
- The exponent of is 2.

Thus, the exponent of is *strictly* greater than the lcm of the exponents of and .