Series-equivalent not implies automorphic in finite abelian group
In terms of subgroups
There can exist a finite abelian group and subgroups and of such that and are series-equivalent subgroups (in other words, is isomorphic to and the quotient group is isomorphic to the quotient group ) but are not automorphic subgroups (i.e., there is no automorphism of sending to ).
The smallest example for has order , and a similar generic example can be constructed for for any prime number .
In terms of extensions
There can be a pair of finite abelian groups and and two extensions with normal subgroup and quotient group such that:
- The total groups in both extensions are abelian, and are isomorphic groups.
- The two extensions are not pseudo-congruent extensions, i.e., they cannot be realized as equivalent to each other using automorphisms of and .
In terms of cohomology and automorphisms
There can be a pair of finite abelian groups and and two elements are elements in the second cohomology group for trivial group action such that:
- and are both represented by symmetric 2-cocycles, hence correspond to abelian group extensions.
- The total groups of the group extensions obtained using the elements and are isomorphic as groups.
- and are not in the same orbit of under the action of .
Equivalence of formulations
- Between extensions and subgroups formulations: The formulation in terms of extensions can be interpreted in terms of subgroups as follows: in the first extension is realized as and as , and in the second extension, is realized as and as . The absence of an automorphism sending to is equivalent to the absence of a pseudo-congruence of extensions.
- Between cohomology and extensions formulations: Direct from the interpretation of the second cohomology group in terms of group extensions.
Here are some intermediate versions:
|Statement||Constraint on||Smallest order of among known examples||Isomorphism class of||Isomorphism class of||Isomorphism class of quotient group|
|series-equivalent abelian-quotient abelian not implies automorphic||and are both abelian||16||nontrivial semidirect product of Z4 and Z4||direct product of Z4 and Z2||cyclic group:Z2|
|series-equivalent characteristic central subgroups may be distinct||and are both central subgroups of||32||SmallGroup(32,28)||cyclic group:Z2||direct product of D8 and Z2|
|series-equivalent abelian-quotient central subgroups not implies automorphic||and are central and are abelian||64||semidirect product of Z8 and Z8 of M-type||direct product of Z4 and Z2||direct product of Z4 and Z2|
The notion of Hall polynomials
Further information: Hall polynomial
Hall polynomials are polynomials that give a formula for the number of subgroups in an abelian group of prime power order having a particular isomorphism class with a particular isomorphism class for the quotient group.
Example of order
We construct an example of an abelian group of order , and subgroups and of order such that and .
We denote by the group of integers modulo .
We define the subgroups and as follows.
Note that the letter used in the definition of should be considered as an integer rather than an integer mod , because its use for the first coordinate requires considering it mod .
Then, and are both of type , and the quotients and are both of type . Thus, and .
However, there is no automorphism of sending to . For this, note that contains elements that are times elements of order , but does not contain any such element.
Note on dual example
Since subgroup lattice and quotient lattice of finite abelian group are isomorphic, we can invert the above example so as to get both and of type and both and of type .