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.