Corrigendum and addendum to: ‘‘Classification of finite groups with all elements of prime order” [Proc.\ Amer.\ Math.\ Soc.\ {\bf 106} (1989), no.\ 3, 625–629; MR0969518 (89k:20038)] by Deaconescu

Cheng, Kai N.; Deaconescu, Marian; Lang, Mong-Lung; Shi, Wu Jie

doi:10.1090/s0002-9939-1993-1116270-7

Cited by 14 publications

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“…A group is a P-group [8] if every nonidentity element of the group has prime order. In this section, we give a lower bound for the λ-number of the power graph of a finite group and compute the exact value of λ( G ) if G is a dihedral group, a generalized quaternion group or a P-group.…”

Section: Lower Boundmentioning

confidence: 99%

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Lambda number of the power graph of a finite group

Feng

Wang

2020

J Algebr Comb

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The power graph G of a finite group G is the graph with the vertex set G, where two distinct elements are adjacent if and only if one is a power of the other. An L(2, 1)-labeling of a graph is an assignment of labels from nonnegative integers to all vertices of such that vertices at distance two get different labels and adjacent vertices get labels that are at least 2 apart. The lambda number of , denoted by λ( ), is the minimum span or range over all L(2, 1)-labelings of . In this paper, we obtain bounds for λ( G ) and give necessary and sufficient conditions when the bounds are attained. As applications, we compute the exact value of λ( G ) if G is a dihedral group, a generalized quaternion group, a P-group or a cyclic group of order pq n , where p and q are distinct primes and n is a positive integer.

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Section: Lower Boundmentioning

confidence: 99%

“…In order to get the desired result, it suffices to show that (5) holds. Now by [8,Main Theorem], one of the following cases occurs:…”

Section: Proofmentioning

confidence: 99%

Lambda number of the power graph of a finite group

Feng

Wang

2020

J Algebr Comb

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“…A similar article was published five years later in the Proceedings of the American Mathematical Society [42], but the main theorem of that article contains some mistakes. In response, the author, together with coauthors, published article [39] as a corrigendum to article [42]. Initially published in Chinese as well, article [185] gained recent attention, prompting the authors of [185] to translate it into English and submited it on arXiv.…”

Section: Introductionmentioning

confidence: 99%

Pure quantitative characterization of finite simple groups

Shi¹

2007

Front. Math. China

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Let G be a finite group, and let π e (G) be the set of all element orders of G. In this short paper we prove that π e (B n (q)) = π e (C n (q)) for all odd q.Keywords finite group, element orders, Lie type simple group MSC 20D06, 20D60Let G be a finite group, and let π e (G) be the set of all element orders of G. Twenty years ago the author of this paper put forward the following conjecture [1]: Conjecture Let G be a group, and let M be a finite simple group. Then G ∼ = M if and only if (i) π e (G) = π e (M ), and (ii) |G| = |M |.For the above conjecture, Professor J. G. Thompson pointed that "This would certainly be a nice theorem" if we prove it. After this, a series of papers proved that this conjecture holds for all sporadic simple groups [1], the alternating groups A n (n 5) [2], the classical simple groups A n (q) [3], 2 A n (q) [4], D n (q) [5,6], 2 D n (q) [5], and all exceptional Lie type simple groups [7,8]. So, the last step of the proof of this conjecture is to prove that the conjecture holds for the simple groups B n (q) and C n (q). Since |B n (q)| = |C n (q)| and B n (q) ∼ = C n (q) for q odd, it is necessary to prove that π e (B n (q)) = π e (C n (q)) for all odd q.First we have the following lemma [9]:Lemma Let h, m be integers and h 2, m 3. If (h, m) = (2, 6), then there exists a prime r such that r | h m − 1, r h i − 1 for i < m. *

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Groups whose elements have given orders

Shi

1997

Chin.Sci.Bull.

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Corrigendum and addendum to: ‘‘Classification of finite groups with all elements of prime order” [Proc.\ Amer.\ Math.\ Soc.\ {\bf 106} (1989), no.\ 3, 625–629; MR0969518 (89k:20038)] by Deaconescu

Abstract: It is shown that the groups in question are either p-groups of exponent p or Frobenius groups of particular type, or they are isomorphic to the simple group A5 ; the misprints and mistakes of a previous paper of the second author are corrected.

Cited by 14 publications

References 5 publications

Lambda number of the power graph of a finite group

Lambda number of the power graph of a finite group

Pure quantitative characterization of finite simple groups

Groups whose elements have given orders

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