# Series-equivalent not implies automorphic in finite abelian group

## Statement

### 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:

1. The total groups in both extensions are abelian, and are isomorphic groups.
2. 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:

1.  and  are both represented by symmetric 2-cocycles, hence correspond to abelian group extensions.
2. The total groups of the group extensions obtained using the elements  and  are isomorphic as groups.
3.  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.

## Related facts

### Weaker facts

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.

## Proof

### 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.

.

.

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 .