# Series-equivalent not implies automorphic

## Statement

It is possible to have a group $G$ and normal subgroups $H$ and $K$ of $G$ that are Series-equivalent subgroups (?) in the sense that $H \cong K$ and $G/H \cong G/K$, but $H$ and $K$ are not automorphic subgroups -- in other words, there is no automorphism of $G$ that sends $H$ to $K$.

## Related facts

### Stronger facts

There are many slight strengthenings of the result that are presented below, along with the smallest order of known examples.

Statement Constraint on $G,H,K$ Smallest order of $G$ among known examples Isomorphism class of $G$ Isomorphism class of $H,K$ Isomorphism class of quotient group $G/H, G/K$
series-equivalent abelian-quotient abelian not implies automorphic $H$ and $G/H$ are both abelian 16 nontrivial semidirect product of Z4 and Z4 direct product of Z4 and Z2 cyclic group:Z2
series-equivalent characteristic central subgroups may be distinct $H$ and $K$ are both central subgroups of $G$ 32 SmallGroup(32,28) cyclic group:Z2 direct product of D8 and Z2
series-equivalent abelian-quotient central subgroups not implies automorphic $H$ and $K$ are central and $G/H, G/K$ are abelian 64 semidirect product of Z8 and Z8 of M-type direct product of Z4 and Z2 direct product of Z4 and Z2
series-equivalent not implies automorphic in finite abelian group $G$ is a finite abelian group 128 direct product of Z8 and Z4 and V4 direct product of Z8 and V4 direct product of Z4 and Z2
characteristic maximal not implies isomorph-free in group of prime power order $H$ and $K$ are maximal, $H$ is characteristic, and $G$ is a group of prime power order 16 nontrivial semidirect product of Z4 and Z4 direct product of Z4 and Z2 cyclic group:Z2
characteristic maximal subgroups may be isomorphic and distinct in group of prime power order Both $H$ and $K$ are characteristic and maximal and $G$ is a group of prime power order 64 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

## Proof

For the proof, see any of the stronger facts listed above.