Characteristicity does not satisfy lower central series condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) not satisfying a subgroup metaproperty (i.e., lower central series condition).
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It is possible to have a group and a characteristic subgroup of such that there is a positive integer for which the lower central series member is not a characteristic subgroup of the lower central series member .
In fact, we can construct, for each positive integer , an example that works for that . In the special case that , we obtain an example of a group and a characteristic subgroup of such that the derived subgroup is not a characteristic subgroup of the derived subgroup .
Example for the derived subgroup ()
Smallest order example for a 2-group
Further information: faithful semidirect product of E8 and Z4
Suppose is the faithful semidirect product of E8 and Z4. We can think of it as the semidirect product of elementary abelian group:E8 (a three-dimensional vector space over field:F2) and cyclic group:Z4, where the generator of the latter acts as the following matrix on the former:
- , the derived subgroup of , is a Klein four-group inside the base of the semidirect product, generated by the first two basis vectors.
- Let . contains the base of the semidirect product as well as the square of the generator of cyclic group:Z4. Explicitly, is the semidirect product of the base by the element acting as follows:
- is isomorphic to direct product of D8 and Z2. The derived subgroup of is isomorphic to cyclic group:Z2 (generated by the second basis vector) and it lives inside as Z2 in V4. Thus, is not a characteristic subgroup of .
Smallest order example for a 3-group
Further information: wreath product of Z3 and Z3
Let be the group . Explicitly, is the semidirect product of elementary abelian group:E27, viewed as a three-dimensional vector space over field:F3, with the acting element of order three acting by the matrix:
- is the subgroup in the base comprising the elements whose coordinates add up to zero. It is an elementary abelian group of prime-square order.
- is defined as the internal semidirect product of and any element of three outside the base. There is a unique possibility for such a subgroup , and it is isomorphic to unitriangular matrix group:UT(3,3).
- is a subgroup of order 3. It lives as a group of order 3 inside which is elementary abelian of order 9, hence is not characteristic in .
Example for larger
Further information: wreath product of groups of prime order
For a positive integer , choose to be a prime number greater than .
- Define as the wreath product of groups of prime order . This is a group of order with an elementary abelian normal subgroup of order and a complement of order acting by cyclic permutation of the coordinates.
- Define as the internal semidirect product of with an element of order outside the elementary abelian normal subgroup.
- and are both elementary abelian subgroups of , with having order and having order . In particular, is a proper nontrivial subgroup of the elementary abelian -group , hence it is not characteristic inside .