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

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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:

## 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.