# Series-equivalent not implies automorphic in finite abelian group

## Statement

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 ).

## Related facts

### 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

We construct an example of an abelian group of order , and subgroups and of order such that and .

We denote by the cyclic group of order .

.

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.