Normality is not transitive for any pair of nontrivial quotient groups

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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

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.