# Endomorphism kernel is not transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., endomorphism kernel)notsatisfying a subgroup metaproperty (i.e., transitive subgroup property).

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## Contents

## Statement

It is possible to have groups such that is an endomorphism kernel in , is an endomorphism kernel in , and is *not* an endomorphism kernel in .

## Related facts

### Similar facts about similar properties

- Normality is not transitive
- Complemented normal is not transitive
- Permutably complemented is not transitive

### Opposite facts about similar properties

### Combined facts with other properties

### Related facts about endomorphism kernel

## Proof

`Further information: quaternion group, subgroup structure of quaternion group`

Suppose is the quaternion group, is a cyclic maximal subgroup, and is the center of quaternion group. Explicitly:

Then:

- is an endomorphism kernel in : In fact, is isomorphic to .
- is an endomorphism kernel in : In fact, is isomorphic to .
- is not an endomorphism kernel in : In fact, is isomorphic to the Klein four-group which does not occur as a subgroup of .