Non-abelian groups in which every subgroup is abelian

Miller, G. A.; Moreno, H. C.

doi:10.1090/s0002-9947-1903-1500650-9

Cited by 180 publications

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“…Z½G=N denote the natural epimorphism and let v ¼ 1 þ ð1 À bÞaB B A Z½G=N be given with B ¼ hbi and with ðv; v Ã Þ a free pair. Choose x; y A G so that yðxÞ ¼ a and yðyÞ ¼ b, and set This motivates the following s-generalization of a result of [7] on minimal nonabelian groups. We have opted to limit the group-theoretic work here by not o¤ering a complete characterization below.…”

Section: Is Central the Results Now Follows Immediately From Theorem mentioning

confidence: 99%

Involutions and free pairs of bicyclic units in integral group rings

Gonçalves¹,

Passman²

2010

Journal of Group Theory

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Abstract. IfÃ : G ! G is an involution on the finite group G, then Ã extends to an involution on the integral group ring Z½G. In this paper, we consider whether bicyclic units u A Z½G exist with the property that the group hu; u Ã i generated by u and u Ã is free on the two generators. If this occurs, we say that ðu; u Ã Þ is a free bicyclic pair. It turns out that the existence of u depends strongly upon the structure of G and on the nature of the involution. One positive result here is that if G is a nonabelian group with all Sylow subgroups abelian, then for any involution Ã , Z½G contains a free bicyclic pair. Free pairs of bicyclic unitsLet R be a commutative ring with 1 and let R½G denote the group ring of G over R.For the most part, we will be concerned with integral group rings where R ¼ Z, or with group algebras where R is a field. Furthermore, we will mainly be interested in finite groups G, although some of our results do hold more generally. If B is a finite subgroup of G, we letB B A R½G denote the sum of the elements of B in R½G. Since ð1 À bÞB B ¼B Bð1 À bÞ ¼ 0 for any b A B, we see that group ring elements of the form ð1 À bÞaB B, with a A G, have square 0. Hence 1 þ ð1 À bÞaB B is a unit in the ring R½G with inverse 1 À ð1 À bÞaB B. When B ¼ hbi is cyclic, elements of the form u ¼ 1 þ ð1 À bÞaB B are known as bicyclic units. It is easy to see that 1 þ ð1 À bÞaB B ¼ 1 if and only if b a ¼ a À1 ba A B and hence if and only if a A N G ðBÞ, the normalizer of B. In particular, if G is a Dedekind group, namely a group with all subgroups normal, then R½G has no nontrivial bicyclic units. Now suppose that Ã : G ! G is an involution, that is, an antiautomorphism of order 2. Then Ã extends to an involution of R½G. In particular, if u is a unit of R½G, then so is u Ã , and we are interested in the nature of the subgroup hu; u Ã i of the unit group that is generated by these two elements. It was shown in [6] that if

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Section: Is Central the Results Now Follows Immediately From Theorem mentioning

confidence: 99%

Involutions and free pairs of bicyclic units in integral group rings

Gonçalves¹,

Passman²

2010

Journal of Group Theory

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“…Thus there is a λ such that λ ≡ 1(mod q), and A p = λI (where I is the identity matrix) and λ p ≡ 1(mod q). That is, G is of type (7).…”

Section: Minimal Non-sqn S-groupsmentioning

confidence: 99%

“…A minimal non-P-group is a non-P-group all of whose proper subgroups are P-groups. The problem of determining all the finite minimal non-P-groups has been studied by several authors and there are many remarkable examples about the minimal non-P-groups: minimal non-abelian groups (Miller and Moreno [7]), minimal non-nilpotent groups (Schmidt), minimal non-supersolvable groups ( [1]) and minimal non-p-nilpotent groups (Itô).…”

Section: Introductionmentioning

confidence: 99%

Finite Groups Which Are Minimal With Respect to S-Quasinormality and Self-Normality

Han¹,

Shi²,

Zhou³

2013

Bulletin of the Korean Mathematical Society

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“…Theorem 1 (Miller-Moreno, [2]). A minimal nonabelian finite 2-group is isomorphic to some of the groups:…”

Section: Introductionmentioning

confidence: 99%

A class of nonabelian nonmetacyclic finite 2-groups

Ćepulić¹

2006

Glas. Mat. Ser. III

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Abstract. Nonabelian nonmetacyclic finite 2-groups in which every proper subgroup is abelian or metacyclic and possessing at least one nonabelian and at least one nonmetacyclic proper subgroup have been investigated and classified. Using the obtained result and two previously known results one gets the complete classification of all nonabelian nonmetacyclic finite 2-groups in which every proper subgroup is abelian or metacyclic.

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Non-abelian groups in which every subgroup is abelian

Cited by 180 publications

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Involutions and free pairs of bicyclic units in integral group rings

Involutions and free pairs of bicyclic units in integral group rings

Finite Groups Which Are Minimal With Respect to S-Quasinormality and Self-Normality

A class of nonabelian nonmetacyclic finite 2-groups

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