Semidirect product is not left-cancellative for finite groups
- 1 Statement
- 2 Related facts
- 3 Proof
Statement from the external semidirect product viewpoint
It is possible to have finite groups , and semidirect products, such that:
but is not isomorphic to .
Statement from the internal semidirect product viewpoint
We can construct a finite group with subgroups such that:
where is isomorphic to but is not isomorphic to .
Related facts about direct products and other notions of products and extensions
- Complement to normal subgroup is isomorphic to quotient
- Complements to normal subgroup are automorphic
- Retract not implies normal complements are isomorphic
Example involving the upper triangular matrices
Suppose is any prime, and let be the group of upper-triangular unipotent matrices over the field of elements. Let be the subgroup of comprising those matrices where the entry is zero. Then, is a group of order .
By fact (1), we have that has two subgroups that are Abelian of maximum order: the top right rectangle groups of dimensions and respectively. Call these subgroups and respectively. Then, observe that:
- Both and are also Abelian subgroups of maximum order in . Moreover, they are the only Abelian subgroups of maximum order in since they are the only Abelian subgroups of maximum order in .
- and are isomorphic -- in fact, they are conjugate subgroups inside the bigger group . This conjugation restricts to an automorphism of , but not of .
- Both and are normal in , and hence in .
- The subgroup with nonzero entries in the positions is a permutable complement to in . Call this subgroup . Then is an internal semidirect product of and . is isomorphic to the prime-cube order group:U3p.
- The subgroup with nonzero entries in the positions is a permutable complement to in . Call this subgroup . Then, is an internal semidirect product of and . Note that is isomorphic to the elementary Abelian group of order .
- Thus, we have , with but not isomorphic to .