# Series-equivalent not implies automorphic

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## Statement

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

## 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  Smallest order of  among known examples Isomorphism class of  Isomorphism class of  Isomorphism class of quotient group 
series-equivalent abelian-quotient abelian not implies automorphic  and  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  and  are both central subgroups of  32 SmallGroup(32,28) cyclic group:Z2 direct product of D8 and Z2
series-equivalent abelian-quotient central subgroups not implies automorphic  and  are central and  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  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  and  are maximal,  is characteristic, and  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  and  are characteristic and maximal and  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.