Series-equivalent not implies automorphic in finite abelian group
Contents
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:
- 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.
Related facts
Weaker facts
Here are some intermediate versions:
Statement | Constraint on ![]() |
Smallest order of ![]() |
Isomorphism class of ![]() |
Isomorphism class of ![]() |
Isomorphism class of quotient group ![]() |
---|---|---|---|---|---|
series-equivalent abelian-quotient abelian not implies automorphic | ![]() ![]() |
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 | ![]() ![]() ![]() |
32 | SmallGroup(32,28) | cyclic group:Z2 | direct product of D8 and Z2 |
series-equivalent abelian-quotient central subgroups not implies automorphic | ![]() ![]() ![]() |
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.
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
.