Series-equivalent not implies automorphic: Difference between revisions

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==Statement==
==Statement==


It is possible to have a [[group]] <math>G</math> and [[normal subgroup]]s <math>H</math> and <math>K</math> of <math>G</math> that are [[fact about::series-equivalent subgroup]]s in the sense that <math>H \cong K</math> and <math>G/H \cong G/K</math>, but <math>H</math> and <math>K</math> are not [[automorphic subgroups]] -- in other words, there is no [[automorphism]] of <math>G</math> that sends <math>H</math> to <math>K</math>.
It is possible to have a [[group]] <math>G</math> and [[normal subgroup]]s <math>H</math> and <math>K</math> of <math>G</math> that are [[fact about::series-equivalent subgroups]] in the sense that <math>H \cong K</math> and <math>G/H \cong G/K</math>, but <math>H</math> and <math>K</math> are not [[automorphic subgroups]] -- in other words, there is no [[automorphism]] of <math>G</math> that sends <math>H</math> to <math>K</math>.


==Related facts==
==Related facts==
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| [[Weaker than::series-equivalent characteristic central subgroups may be distinct]] || <math>H</math> and <math>K</math> are both [[central subgroup]]s of <math>G</math> || 32 || [[SmallGroup(32,28)]] || [[cyclic group:Z2]] || [[direct product of D8 and Z2]]
| [[Weaker than::series-equivalent characteristic central subgroups may be distinct]] || <math>H</math> and <math>K</math> are both [[central subgroup]]s of <math>G</math> || 32 || [[SmallGroup(32,28)]] || [[cyclic group:Z2]] || [[direct product of D8 and Z2]]
|-
|-
| [[Weaker than::series-equivalent abelian-quotient central subgroups may be distinct]] || <math>H</math> and <math>K</math> are central and <math>G/H, G/K</math> are distinct || 64 || [[semidirect product of Z8 and Z8 of M-type]] || [[direct product of Z4 and Z2]] || [[direct product of Z4 and Z2]]
| [[Weaker than::series-equivalent abelian-quotient central subgroups not implies automorphic]] || <math>H</math> and <math>K</math> are central and <math>G/H, G/K</math> are abelian || 64 || [[semidirect product of Z8 and Z8 of M-type]] || [[direct product of Z4 and Z2]] || [[direct product of Z4 and Z2]]
|-
|-
| [[Weaker than::series-equivalent not implies automorphic in finite abelian group]] || <math>G</math> 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]]
| [[Weaker than::series-equivalent not implies automorphic in finite abelian group]] || <math>G</math> 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]]
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| [[Weaker than::characteristic maximal subgroups may be isomorphic and distinct in group of prime power order]] || Both <math>H</math> and <math>K</math> are characteristic and maximal and <math>G</math> is a [[group of prime power order]] || 64 || {{fillin}} || {{fillin}} || {{fillin}}
| [[Weaker than::characteristic maximal subgroups may be isomorphic and distinct in group of prime power order]] || Both <math>H</math> and <math>K</math> are characteristic and maximal and <math>G</math> is a [[group of prime power order]] || 64 || {{fillin}} || {{fillin}} || {{fillin}}
|}
|}
==Proof==
For the proof, see any of the stronger facts listed above.

Latest revision as of 23:06, 1 February 2011

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≅K and G/H≅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.

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