The full automorphism group of the power (di)graph of a finite group

Feng, Min; Ma, Xuanlong; Wang, Kaishun

doi:10.1016/j.ejc.2015.10.006

Cited by 33 publications

(27 citation statements)

References 10 publications

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“…Hence, the conjecture does not hold if n = p m for any prime p and integer m > 2. In [17], proved that this conjecture holds for the remaining case. Feng, Ma and Wang [17], describe the full automorphism group of the power (di)graph of an arbitrary finite group.…”

Section: Automorphism Groups Of Power Graphsmentioning

confidence: 87%

A Survey on the Automorphism Groups of the Commuting Graphs and Power Graphs

Mirzargar¹

2019

FU Math Inform

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Let G be a nite group. The power graph P(G) of a group G is the graphwhose vertex set is the group elements and two elements are adjacent if one is a power of the other. The commuting graph \Delta(G) of a group G, is the graph whose vertices are the group elements, two of them joined if they commute. When the vertex set is G-Z(G), this graph is denoted by \Gamma(G). Since the results based on the automorphism group of these kinds of graphs are so sporadic, in this paper, we give a survey of all results on the automorphism group of power graphs and commuting graphs obtained in the literature.

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Section: Automorphism Groups Of Power Graphsmentioning

confidence: 87%

A Survey on the Automorphism Groups of the Commuting Graphs and Power Graphs

Mirzargar¹

2019

FU Math Inform

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“…we deduce that the types [1 4 , 3 2 ], [1 3 , 7], [3,7], [1 7 , 3], [1 2 , 3, 5] and [1 5 , 5] are admissible for Ω 10 . Since [10] / ∈ T (A 10 ), the vertices of type T 2 = [5 2 ] ∈ T (A 10 ) are instead isolated and thus c 0 (A 10 ) T2 = µ [5 2 ] [A 10 ] = 18144. The same argument used for P 0 (A 9 ) shows also that for T 3 = [1, 3 3 ] we have c 0 (A 10 ) T3 = µ [1,3 3 ] [A 10 ] = 11200.…”

Section: Higher Degreesmentioning

confidence: 99%

“…For instance, in [11] it is shown that the power graph has a transitive orientation and a closed formula for the metric dimension of P (G) is established. In [10], relying on a fundamental result about groups with isomorphic power graphs ([6, Proposition 1]), the full automorphism group of P (G) is described. In many cases such as [8,15,14,9,12], those properties are investigated in relation to the group theoretical properties of G. In particular, the groups G for which P (G) is planar or toroidal or projective are classified in [9]; in [12] it is presented a characterization of the chromatic number of P (G) and the groups whose power graphs are uniquely colorable, split or unicyclic are classified.…”

Section: Introductionmentioning

confidence: 99%

On some graphs associated with the finite alternating groups

Bubboloni

Iranmanesh

Shaker

2017

Communications in Algebra

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Let P 0 (An), P 0 (An), P 0 (T (An)) and O 0 (An) be respectively the proper power graph, the proper quotient power graph, the proper power type graph and the proper order graph of the alternating group An, for n ≥ 3. We determine the number of the components of those graphs. In particular, we prove that the power graph P (An) is 2-connected if and only if the power type graph P (T (An)) is 2-connected, if and only if either n = 3 or none of n, n − 1, n − 2, n 2 and n−1 2 is a prime. We also give some information on the properties of those components. MSC(2010): Primary: 05C25; Secondary: 20B30.

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“…Hence, the conjecture does not hold if n = p m . In June 2014, Min Feng, Xuanlong Ma, Kaishun Wang [5] proved that the conjecture holds for the remaining cases, that is for n = p m . In fact they proved a more general result, but their proof uses some what complicated group theoritic arguments.…”

Section: Introductionmentioning

confidence: 98%