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

## 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 .

## 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.