Prime graph components of finite groups

Williams, Julian

doi:10.1016/0021-8693(81)90218-0

Cited by 462 publications

(367 citation statements)

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“…As a generalization of this result, we prove the following theorem: By using the classification of finite simple groups, the structure of a finite group G such that its prime graph is not connected has been determined by Gruenberg and Kegel, in an unpublished paper. Later, Williams published this result together with a classification of finite simple groups with disconnected prime graph, which are distinct from Lie-type groups of even characteristic; see [24]. In [8], a similar description was given for simple Lie-type groups in an even characteristic.…”

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confidence: 77%

On r-recognition by prime graph of B p (3) where p is an odd prime

Momen

Khosravi

2010

Monatsh Math

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confidence: 77%

On r-recognition by prime graph of B p (3) where p is an odd prime

Momen

Khosravi

2010

Monatsh Math

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“…According to this partition, ω i (G) is a subset of π i (G)-numbers of ω(G) for every 1 i s(G). LEMMA 2.1 (Gruenberg-Kegel, see [24]). If G is a finite group with s(G) > 1, then one of the following holds:…”

Section: Gruenberg-kegel Graph and Groups Isospectral To Simple Groupsmentioning

confidence: 98%

Characterization of the finite simple groups by spectrum and order

2009

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We give an affirmative answer to Question 12.39 in the Kourovka Notebook. Namely, it is proved that a finite simple group and a finite group having equal orders and same sets of element orders are isomorphic.The spectrum of a group G is the set ω(G) of its element orders. The spectrum of a finite group G together with its order retains a substantial part of information on the structure of G but, as demonstrated by the example of the dihedral group D 8 of order 8 and the quaternion group Q 8 , does not necessarily determine G uniquely. In [1], W. Shi conjectured that the desired uniqueness would be achieved for finite simple groups. In [2, Question 12.39], A. S. Kondratiev worded this conjecture as follows:Is it true that a finite group and a finite simple group are isomorphic if they have equal orders and same sets of element orders?Later, for brevity, it was suggested to refer to a finite group G that is isomorphic to every finite group H with ω(H) = ω(G) and |H| = |G| as recognizable by spectrum and order. In this terminology, Shi's question reads as follows:Is it true that all finite simple groups are recognizable by spectrum and order?The answer to this question is obviously 'yes' for Abelian simple groups. In a series of papers [1,[3][4][5][6][7][8], an affirmative answer was given for all non-Abelian simple groups except the symplectic groups, orthogonal groups of odd dimension, and orthogonal groups of type D n with n even. In the present paper, we argue for the recognizability by spectrum and order for these remaining groups. Thus, Shi's conjecture is confirmed and the following theorem holds true.

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“…By Lemma 3.3, w 0 is of p-defect zero. Recall that p is adjacent to 2 (see [26]). Suppose that p ¼ 3.…”

Section: Lemma 32 ([23mentioning

confidence: 99%