# Elementary abelian-to-normal replacement fails for Klein four-group

This article discusses a failure of replacement, i.e., a situation where the analogue of a valid replacement theorem fails to hold under slightly modified conditions.

View other failures of replacement | View replacement theorems

## Contents

## Statement

### Hands-on statement

We can find a finite group whose order is a power of , that contains a Klein four-group (?) as a subgroup(i.e., it contains an elementary abelian subgroup of order four) but does not contain any normal subgroup that is a Klein four-group.

In fact, the following stronger statement is true: for any , we can find a finite group whose order is a power of , that contains a Klein four-subgroup but does not contain a -subnormal Klein four-subgroup.

### Statement in terms of the weak normal replacement condition

The single-element collection comprising the Klein four-group is *not* a Collection of groups satisfying a weak normal replacement condition (?) for the prime .

## Related facts

### Similar facts

### Opposite facts

- Elementary abelian-to-normal replacement theorem for prime-square order: The statement that fails for is true for odd primes.
- Abelian-to-normal replacement theorem for prime-cube order
- Jonah-Konvisser elementary abelian-to-normal replacement theorem
- Jonah-Konvisser abelian-to-normal replacement theorem

## Proof

### Example of the dihedral group (for the weaker version)

`Further information: dihedral group:D16, Subgroup structure of dihedral group:D16`

Consider the dihedral group of order sixteen:

.

Then:

- The subgroup is a Klein four-group.
- There is no normal Klein four-subgroup: Any subgroup contained in is cyclic, so it cannot be a Klein four-group. Thus, a normal Klein four-subgroup, if it exists, must contain some element outside . However, the sizes of the conjugacy classes of elements outside is each, so any normal subgroup containing an element outside must have order strictly bigger than four.

### Example of the dihedral group (for the stronger version)

`Further information: Dihedral group, Subgroup structure of dihedral groups`

We need to take a dihedral group of order , as follows:

.

Then:

- The subgroup is a Klein four-group.
- There is no -subnormal Klein four-subgroup: Any subgroup contained in is cyclic, so it cannot be a Klein four-group. Thus, a -subnormal Klein four-subgroup, if it exists, must contain some element outside . Since all elements outside are in the same automorphism class, we can assume that it contains . However, the smallest -subnormal subgroup containing , computed by taking the normal closure times, is a dihedral group of order eight given by .