Characteristic maximal subgroups may be isomorphic and distinct in group of prime power order

From Groupprops
Jump to: navigation, search

Statement

It is possible to have a group of prime power order G and two subgroups H and K of G such that:

  1. H and K are both Characteristic subgroup (?)s of G.
  2. H and K are both maximal subgroups of G.
  3. H and K are isomorphic groups: They are hence also Series-equivalent subgroups (?) of G, because the quotients G/H and G/K are both cyclic of the same prime order.
  4. H and K are distinct subgroups, i.e., they are not equal as subgroups of G.

Related facts

See series-equivalent not implies automorphic#Related facts.

Proof

The smallest known example is with G a group of order 64. There are many examples with this order, one of which has G isomorphic to SmallGroup(64,13) and H and K both isomorphic to nontrivial semidirect product of Z4 and Z8 (group ID: (32,12)).