Series-equivalent not implies automorphic in finite abelian group
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 ).
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.
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.