Series-equivalent not implies automorphic: Difference between revisions

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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Proof

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

@@ Line 16: / Line 16: @@
 | [[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]]