On finite groups with disconnected prime graph

Zinov’eva, M. R.; Mazurov, V. D.

doi:10.1134/s0081543813090149

Cited by 8 publications

(3 citation statements)

References 11 publications

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“…We recall that a group G is called a 2-Frobenius group if G = ABC, where A and AB are normal subgroups of G, B is a normal subgroup of BC, and AB and BC are Frobenius groups. Zinovèva and V. D. Mazurov observed that the prime graph of a 2-Frobenius group is always disconnected, more precisely, it is the union of two connected components each of which is a complete graph [42,Lemma 3(a) ]. On the other hand, Mazurov constructed a 2-Frobenius group of the same order as the simple group U 4 (2) ( [20,42]).…”

Section: G Condition On G H Od (G) Referencesmentioning

confidence: 99%

“…Zinovèva and V. D. Mazurov observed that the prime graph of a 2-Frobenius group is always disconnected, more precisely, it is the union of two connected components each of which is a complete graph [42,Lemma 3(a) ]. On the other hand, Mazurov constructed a 2-Frobenius group of the same order as the simple group U 4 (2) ( [20,42]). In particular, this shows that h OD (U 4 (2)) 2 (see also [32]).…”

Section: G Condition On G H Od (G) Referencesmentioning

confidence: 99%

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Several Quantitative Characterizations of Some Specific Groups

Mohammadzadeh¹,

Moghaddamfar²

2015

Preprint

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Let G be a finite group and let π(G) = {p 1 , p 2 , . . . , p k } be the set of prime divisors of |G| for which, is defined as follows: its vertex set is π(G) and two different vertices p i and p j are adjacent by an edge if and only if G contains an element of order p i p j . The degree of a vertexis the vertex set of a connected component of GK(G), then the largest ω-number which divides |G|, is said to be an order component of GK(G). We will say that the problem of OD-characterization is solved for a finite group if we find the number of pairwise non-isomorphic finite groups with the same order and degree pattern as the group under study. The purpose of this article is twofold. First, we completely solve the problem of OD-characterization for every finite non-abelian simple group with orders having prime divisors at most 29. In particular, we show that there are exactly two non-isomorphic finite groups with the same order and degree pattern as U 4 (2). Second, we prove that there are exactly two non-isomorphic finite groups with the same order components as U 5 (2).

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Section: G Condition On G H Od (G) Referencesmentioning

confidence: 99%

Section: G Condition On G H Od (G) Referencesmentioning

confidence: 99%

Several Quantitative Characterizations of Some Specific Groups

Mohammadzadeh¹,

Moghaddamfar²

2015

Preprint

View full text Add to dashboard Cite

show abstract

“…Zinov'eva and V.D. Mazurov observed that the prime graph of a 2-Frobenius group is always disconnected, more precisely, it is the union of two connected components each of which is a complete graph [45,Lemma 3(a)]. On the other hand, Mazurov constructed a 2-Frobenius group of the same order as the simple group U 4 (2) ( [20], [44]).…”

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confidence: 99%

Several quantitative characterizations of some specific groups

Mohammadzadeh¹,

Moghaddamfar²

2017

Commentationes Mathematicae Universitatis Carolinae

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A characterization of the prime graphs of solvable groups

et al. 2015

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Let π(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G, denoted Γ G , is the graph with vertex set π(G) with edges {p, q} ∈ E(Γ G ) if and only if there exists an element of order pq in G. In this paper, we prove that a graph is isomorphic to the prime graph of a solvable group if and only if its complement is 3-colorable and triangle free. We then introduce the idea of a minimal prime graph. We prove that there exists an infinite class of solvable groups whose prime graphs are minimal. We prove the 3k-conjecture on prime divisors in element orders for solvable groups with minimal prime graphs, and we show that solvable groups whose prime graphs are minimal have Fitting length 3 or 4.

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On finite groups with disconnected prime graph

Abstract: Abstract-All finite simple nonabelian groups that have the same prime graph as a Frobenius group or a 2-Frobenius group are found.

Cited by 8 publications

References 11 publications

Several Quantitative Characterizations of Some Specific Groups

Several Quantitative Characterizations of Some Specific Groups

Several quantitative characterizations of some specific groups

A characterization of the prime graphs of solvable groups

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