Sum of Element Orders of Maximal Subgroups of the Symmetric Group

Amiri, Habib; Amiri, S. M. Jafarian

doi:10.1080/00927872.2010.537290

Cited by 17 publications

(10 citation statements)

References 6 publications

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“…, 50}. In this regard, we mention that some useful results on the sum of element orders of symmetric groups were proved in [3] and we suggest the following open problem:…”

Section: Restricting To Normal Subgroupsmentioning

confidence: 94%

“…In [19] (see Theorem 1.1), the reader can find a classification of finite groups having all non-trivial proper normal subgroups of the same order. Such a group G is isomorphic to -A simple group; -A group having a unique proper normal subgroup; -S × S, where S is a simple group; -A 8 × P SL (3,4); -B n (q) × C n (q), where n ≥ 3 and q is an odd prime power.…”

Section: Proposition 32 Let G Be a Finite Solvable Group Having A Unique Proper Normal Subgroup Then G Is Not ψ-Normal Divisiblementioning

confidence: 99%

“…As we previously stated, all finite simple groups are ψ-normal divisible. One can use GAP to check that A 8 × P SL (3,4) is not ψ-normal divisible. Further, we focus on finite groups having a unique proper normal subgroup, while the other two remaining cases are open.…”

Section: Proposition 32 Let G Be a Finite Solvable Group Having A Unique Proper Normal Subgroup Then G Is Not ψ-Normal Divisiblementioning

confidence: 99%

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On a Divisibility Property Involving the Sum of Element Orders

Lazorec

2020

Bull. Malays. Math. Sci. Soc.

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A finite group G is called ψ-divisible if ψ(H )|ψ(G) for any subgroup H of G, where ψ(H ) and ψ(G) are the sum of element orders of H and G, respectively. In this paper, we classify the finite groups whose subgroups are all ψ-divisible. Since the existence of ψ-divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the ψ-divisibility property of ZM-groups. In the end, we introduce the concept of ψ-normal divisible group, i.e., a group for which the ψ-divisibility property is satisfied by all its normal subgroups. Using simple and quasisimple groups, we are able to construct infinitely many ψ-normal divisible groups which are neither simple nor nilpotent.

show abstract

“…, 50}. In this regard, we mention that some useful results on the sum of element orders of symmetric groups were proved in [3] and we suggest the following open problem:…”

Section: Restricting To Normal Subgroupsmentioning

confidence: 94%

Section: Proposition 32 Let G Be a Finite Solvable Group Having A Unique Proper Normal Subgroup Then G Is Not ψ-Normal Divisiblementioning

confidence: 99%

Section: Proposition 32 Let G Be a Finite Solvable Group Having A Unique Proper Normal Subgroup Then G Is Not ψ-Normal Divisiblementioning

confidence: 99%

See 1 more Smart Citation

On a Divisibility Property Involving the Sum of Element Orders

Lazorec

2020

Bull. Malays. Math. Sci. Soc.

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“…However, this conjecture has recently been shown to be false [3]. In 2012, H. Amiri and S. M. J. Amiri [4] showed that ( ) < ( ) for all proper subgroups ̸ = of , where is the symmetric group and is the alternating group. In this paper, we give characterizations of various infinite sets of finite groups , given certain restrictions on ( ) or ( ) (or both), where is a proper subgroup of .…”

Section: Introductionmentioning

confidence: 99%

Characterizing Finite Groups Using the Sum of the Orders of the Elements

Harrington

Jones

Lamarche

2014

International Journal of Combinatorics

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“…Let G be a finite non-solvable group, which contains no non-trivial solvable normal subgroups and let θ ∈ Aut(G). If θ inverts more than 2 9 |G| of the elements of G, then G ≃ A 5 .…”

mentioning

confidence: 99%