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

This article gives the statement, and possibly proof, of a group property (i.e., finite abelian group) not satisfying 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) not satisfying 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) not satisfying 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 $G_{1},G_{2}$ such that:

• The lattice of subgroups of $G_{1}$ is isomorphic to the lattice of subgroups of $G_{2}$
• $G_{1}$ is a finite abelian group, finite nilpotent group, and group of prime power order
• $G_{2}$ is not a finite abelian group, finite nilpotent group, or group of prime power order.

Related facts

• No subgroup property between normal Sylow and subnormal or between Sylow retract and retract is conditionally lattice-determined

Proof

Choose primes $p,q$ such that $q$ divides $p-1$. Choose the following:

• $G_{1}$ is the elementary abelian group of order $p^{2}$. It is clearly finite abelian, finite nilpotent, and of prime power order..
• $G_{2}$ is the external semidirect product of the cyclic group of order $p$ by the cyclic subgroup of order $q$ in its automorphism group. It is clearly non-abelian and not of prime power order (its order is $pq$).

Both $G_{1}$ and $G_{2}$ have a lattice of size $p+3$, including the trivial subgroup, whole group, and $p+1$ intermediate, mutually incomparable subgroups. These lattices are clearly isomorphic.

The smallest example is $p=3,q=2$, giving $G_{1}$ as elementary abelian group:E9 and $G_{2}$ as symmetric group:S3.