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Groupprops β

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

Statement

It is possible to have two groups G_1, G_2 such that:

Related facts

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.