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1
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OnCSQ-normal subgroups of finite groups

100%
Xu Y., Li X.
Open Mathematics
|
2016
|
tom 14
|
nr 1
801-806
EN
We introduce a new subgroup embedding property of finite groups called CSQ-normality of subgroups. Using this subgroup property, we determine the structure of finite groups with some CSQ-normal subgroups of Sylow subgroups. As an application of our results, some recent results are generalized.
2
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On a class of finite solvable groups

100%
Beidleman J., Heineken H., Schmidt J.
Open Mathematics
|
2013
|
tom 11
|
nr 9
1598-1604
EN
A finite solvable group G is called an X-group if the subnormal subgroups of G permute with all the system normalizers of G. It is our purpose here to determine some of the properties of X-groups. Subgroups and quotient groups of X-groups are X-groups. Let M and N be normal subgroups of a group G of relatively prime order. If G/M and G/N are X-groups, then G is also an X-group. Let the nilpotent residual L of G be abelian. Then G is an X-group if and only if G acts by conjugation on L as a group of power automorphisms.
3
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Finite groups whose all second maximal subgroups are cyclic

100%
Ma L., Meng W., Ma W.
Open Mathematics
|
2017
|
tom 15
|
nr 1
611-615
EN
In this paper, we give a complete classification of the finite groups G whose second maximal subgroups are cyclic
4
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Algorithms for permutability in finite groups

76%
Ballester-Bolinches A., Cosme-Llópez E., Esteban-Romero R.
Open Mathematics
|
2013
|
tom 11
|
nr 11
1914-1922
EN
In this paper we describe some algorithms to identify permutable and Sylow-permutable subgroups of finite groups, Dedekind and Iwasawa finite groups, and finite T-groups (groups in which normality is transitive), PT-groups (groups in which permutability is transitive), and PST-groups (groups in which Sylow permutability is transitive). These algorithms have been implemented in a package for the computer algebra system GAP.
5
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Groups where each element is conjugate to its certain power

76%
Hegedűs P.
Open Mathematics
|
2013
|
tom 11
|
nr 10
1742-1749
EN
This paper deals with a rationality condition for groups. Let n be a fixed positive integer. Suppose every element g of the finite solvable group is conjugate to its nth power g n. Let p be a prime divisor of the order of the group. We conclude that the multiplicative order of n modulo p is small, or p is small.
6
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Erratum to: “Subnormal, permutable, and embedded subgroups in finite groups”

76%
Beidleman J., Ragland M.
Open Mathematics
|
2012
|
tom 10
|
nr 2
835-836
EN
The original version of the article was published in Central European Journal of Mathematics, 2011, 9(4), 915–921, DOI: 10.2478/s11533-011-0029-8. Unfortunately, the original version of this article contains a mistake: Lemma 2.1 (2) is not true. We correct Lemma 2.2 (2) and Theorem 1.1 in our paper where this lemma was used.
7
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Maximal subgroups and PST-groups

64%
Ballester-Bolinches A., Beidleman J., Esteban-Romero R., Pérez-Calabuig V.
Open Mathematics
|
2013
|
tom 11
|
nr 6
1078-1082
EN
A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19–25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versions of Kaplan’s results, which enables a better understanding of the relationships between these classes.
8
Content available remote

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

46%
Bandman T., Garion S., Kunyavskiĭ B.
Open Mathematics
|
2014
|
tom 12
|
nr 2
175-211
EN
We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.
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