On capable $p$-groups of nilpotency class two

Bacon, Michael R.; Kappe, Luise-Charlotte

doi:10.1215/ijm/1258488137

Cited by 26 publications

(34 citation statements)

References 6 publications

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“…For every element g ∈ G, the exterior centralizer of g in G, which is denoted by C ∧ G g , is the set of all elements g ∈ G such that g ∧ g = 1 and the exterior centre of G denoted by Z ∧ G is the intersection of all exterior centralizers of elements of G; see Bacon and Kappe [1] for more details. Recall that by Ellis [7], a group G is capable if and only if Z ∧ G = 1.…”

Section: Introductionmentioning

confidence: 99%

On the Exterior Degree of Finite Groups

Niroomand

Rezaei

2010

Communications in Algebra

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We introduce the exterior degree of a finite group G to be the probability for two elements g and g in G such that g ∧ g = 1, and we shall state some results concerning this concept. We show that if G is a non-abelian capable group, then its exterior degree is less than 1/p, where p is the smallest prime number dividing the order of G. Finally, we give some relations between the new concept and commutativity degree, capability, and the Schur multiplier.

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Section: Introductionmentioning

confidence: 99%

On the Exterior Degree of Finite Groups

Niroomand

Rezaei

2010

Communications in Algebra

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“…Capability of groups has received renewed attention in recent years, thanks to results of Beyl, Felgner, and Schmid [5] characterizing the capability of a group in terms of its epicenter, and work of Ellis [6] describing the epicenter in terms of the non-abelian tensor square of the group. The epicenter was used in [5] to characterize the capable extra-special p-groups; and the non-abelian tensor square was used in [1] to characterize the capable 2-generator finite p-groups of odd order and class 2.…”

Section: Introductionmentioning

confidence: 99%

“…At the end of [1], for example, the authors note that their methods require 'very explicit knowledge of the groups' in question.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 5 we characterize the capable 2-nilpotent products of cyclic 2-groups. Then in Section 6 we use our results on capable 2-nilpotent products of cyclic p-groups to derive a characterization of the capable 2-generated nilpotent p-groups of class 2 for p an odd prime (recently obtained through di¤erent methods by Bacon and Kappe [1]), and to prove other related results, to illustrate that our methods may be useful in further investigations.…”

Section: Introductionmentioning

confidence: 99%

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Capability of nilpotent products of cyclic groups

Magidin¹

2005

Journal of Group Theory

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A group is called capable if it is a central factor group. We consider the capability of nilpotent products of cyclic groups, and obtain a generalization of a theorem of Baer for the small class case. The approach is also used to obtain some recent results on the capability of certain nilpotent groups of class 2. We also establish a necessary condition for the capability of an arbitrary p-group of class k, and some further results.

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“…97 (2011) On capable groups of order p 2 q 301 are direct sums of cyclic groups; the capable extra-special p-groups were characterized by Beyl et al [3] (only the dihedral group of order 8 and the extraspecial groups of order p 3 and exponent p are capable); they also described the metacyclic groups which are capable. Magidin [15,16], characterized the 2-generated capable p-groups of class two (for odd p, independently obtained in part by Bacon and Kappe [1]). …”

Section: Introductionmentioning

confidence: 99%