# Normality is not transitive for any pair of nontrivial quotient groups

From Groupprops

## Statement

Suppose and are (possibly equal) nontrivial groups. Then, there exist groups such that all the following conditions are satisfied:

- is a Normal subgroup (?) of and the quotient group is isomorphic to .
- is a normal subgroup of and the quotient group is isomorphic to .
- is not a normal subgroup of .

## Related facts

### Similar facts

- Normality is not transitive
- Conjunction of normality with any nontrivial finite-direct product-closed property of groups is not transitive
- Normality is not transitive for any nontrivially satisfied extension-closed group property
- There exist subgroups of arbitrarily large subnormal depth

### Generalizations

## Proof

The construction is as follows. Let be the wreath product of and for the regular group action of . Let be the subgroup , i.e., the normal subgroup that forms the base of the semidirect product, and let be the subgroup of where a particular coordinate is the identity element. (If we are thinking of as functions from to , then can be taken as the subgroup comprising those functions that send the identity element of to the identity element of -- here, the *particular coordinate* becomes the coordinate corresponding to the identity element of ).

Thus, is isomorphic to . Then:

- is normal in and is isomorphic to :
**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE] - is normal in and is isomorphic to :
**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE] - is not normal in : The action of on the coordinates in is transitive on the coordinates, so is not preserved under this action.