Capability of nilpotent products of cyclic groups

Magidin, Arturo

doi:10.1515/jgth.2005.8.4.431

Cited by 14 publications

(28 citation statements)

References 19 publications

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“…These results follow from [3,12] in the case of odd primes p and from [13] for p = 2. Likewise, the condition G/G capable is not sufficient to imply G capable in the case p = 2.…”

Section: Corollary 43 Let G Be a Two-generator Non-torsion Group Ofsupporting

confidence: 72%

“…According to the necessary condition, all groups with torsion-free rank 1 are not capable and no further calculations of the epicentre are necessary. Thus, our necessary criterion for the non-torsion case is very similar to the one given by Hall [10] and Magidin [12] for capable p-groups of class 2, guaranteeing the existence of at least two elements of maximal order in every generating set.…”

Section: Definition 15 ([7]) For Any Group G the Exterior Square Omentioning

confidence: 94%

“…Explicit knowledge of the tensor square of twogenerator p-groups of class 2, p an odd prime, enabled the authors of [3], to investigate the capability of such groups. In [12], using different methods, Magidin investigated the same class of groups and determined the capable ones among them. In addition, he generalised a necessary condition for the capability of finite p-groups established by Hall in [10].…”

Section: Definition 15 ([7]) For Any Group G the Exterior Square Omentioning

confidence: 99%

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On the Capability of Finitely Generated Non-Torsion Groups of Nilpotency Class 2

Kappe

Ali²,

Sarmin³

2011

Glasgow Math. J.

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Abstract.A group is called capable if it is a central factor group. In this paper, we establish a necessary condition for a finitely generated non-torsion group of nilpotency class 2 to be capable. Using the classification of two-generator non-torsion groups of nilpotency class 2, we determine which of them are capable and which are not and give a necessary and sufficient condition for a two-generator non-torsion group of class 2 to be capable in terms of the torsion-free rank of its factor commutator group.2010 Mathematics Subject Classification. 20D15, 20J05.

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“…These results follow from [3,12] in the case of odd primes p and from [13] for p = 2. Likewise, the condition G/G capable is not sufficient to imply G capable in the case p = 2.…”

Section: Corollary 43 Let G Be a Two-generator Non-torsion Group Ofsupporting

confidence: 72%

Section: Definition 15 ([7]) For Any Group G the Exterior Square Omentioning

confidence: 94%

Section: Definition 15 ([7]) For Any Group G the Exterior Square Omentioning

confidence: 99%

See 1 more Smart Citation

On the Capability of Finitely Generated Non-Torsion Groups of Nilpotency Class 2

Kappe

Ali²,

Sarmin³

2011

Glasgow Math. J.

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“…In this section we will extend the main result from [2] to the case k = p with p an arbitrary prime. We will do so by computing the center of a (p + 1)-nilpotent product of cyclic p-groups much in the same way as above, using a normal form for the elements of such a product that was obtained by R.R.…”

Section: The Case K = Pmentioning

confidence: 89%

“…In [2], the last clause of Lemma 4.2(ii) is incorrect. Because of this error, the last assertion in Lemma 4.3 is also incorrect; the proof of Theorem 4.4, which describes the center of a k-nilpotent product of cyclic p-groups when k ≤ p, relied on that incorrect assertion and so has a gap.…”

Section: Shoving Commutatorsmentioning

confidence: 99%

Capability of nilpotent products of cyclic groups II

Magidin

2007

Journal of Group Theory

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Abstract. In Part I it was shown that if G is a p-group of class k, generated by elements of orders 1 < p α 1 ≤ · · · ≤ p αr , then a necessary condition for the capability of G is that r > 1 and αr ≤ α r−1 + ⌊ k−1 p−1 ⌋. It was also shown that when G is the k-nilpotent product of the cyclic groups generated by those elements and k = p = 2 or k < p, then the given conditions are also sufficient. We make a correction related to the small class case, and extend the sufficiency result to k = p for arbitrary prime p.Recall that a group G is said to be capable if and only if there exists a group H such that G ∼ = H/Z(H), where Z(H) is the center of H. In [2] we proved that if G is a capable p-group of class k, generated by x 1 , . . . , x r , with x i of order p αi , 1 ≤ α 1 ≤ · · · ≤ α r , then r > 1 and α r ≤ α r−1 + ⌊ k−1 p−1 ⌋. We also proved that if G is the k-nilpotent product of the cyclic p-groups generated by the x i , then the conditions are also sufficient for the cases k < p and k = p = 2.The purpose of this note is twofold: first, we will note an error in a lemma that was used in the proof of the small class case and make the necessary corrections to justify that result. Second, we will extend the result to the case k = p with p an arbitrary prime. Since we follow closely on [2], we refer the reader there for the relevant definitions and conventions. I am extremely grateful to Prof. T. C. Hurley who brought to my attention the results from [1,7]; these results allowed the correction of the error noted above, as well as simplifying my argument for the k = p case.

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