# There exists an abelian group of prime power order that is lattice-isomorphic to a non-abelian group not of prime power order

From Groupprops

This article gives the statement, and possibly proof, of a group property (i.e., finite abelian group)notsatisfying a group metaproperty (i.e., lattice-determined group property).

View all group metaproperty dissatisfactions | View all group metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for group properties

Get more facts about finite abelian group|Get more facts about lattice-determined group property|

This article gives the statement, and possibly proof, of a group property (i.e., finite nilpotent group)notsatisfying a group metaproperty (i.e., lattice-determined group property).

View all group metaproperty dissatisfactions | View all group metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for group properties

Get more facts about finite nilpotent group|Get more facts about lattice-determined group property|

This article gives the statement, and possibly proof, of a group property (i.e., group of prime power order)notsatisfying a group metaproperty (i.e., lattice-determined group property).

View all group metaproperty dissatisfactions | View all group metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for group properties

Get more facts about group of prime power order|Get more facts about lattice-determined group property|

## Statement

It is possible to have two groups such that:

- The lattice of subgroups of is isomorphic to the lattice of subgroups of
- is a finite abelian group, finite nilpotent group, and group of prime power order
- is not a finite abelian group, finite nilpotent group, or group of prime power order.

## Related facts

## Proof

Choose primes such that divides . Choose the following:

- is the elementary abelian group of order . It is clearly finite abelian, finite nilpotent, and of prime power order..
- is the external semidirect product of the cyclic group of order by the cyclic subgroup of order in its automorphism group. It is clearly non-abelian and not of prime power order (its order is ).

Both and have a lattice of size , including the trivial subgroup, whole group, and intermediate, mutually incomparable subgroups. These lattices are clearly isomorphic.

The smallest example is , giving as elementary abelian group:E9 and as symmetric group:S3.