A Note on Hobby’s Theorem of Finite Groups

Throughout the paper we consider only finite groups.J. C. Beidleman and H. Smith [3] have proposed the following question: “If G is a group and Ha subnormal subgroup of G containing Φ(G), the Frattini subgroup of G, such that H/Φ(G)is supersoluble, is H necessarily supersoluble? “In this paper, we give not only an affirmative answer to this question but also we see that the above result still holds if supersoluble is replaced by any saturated formation containing the class of all nilpotent groups.

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Impartial achievement games for generating nilpotent groups

Journal of Group Theory ◽

10.1515/jgth-2018-0117 ◽

2019 ◽

Vol 22 (3) ◽

pp. 515-527

Author(s):

Bret J. Benesh ◽

Dana C. Ernst ◽

Nándor Sieben

Keyword(s):

Finite Group ◽

Finite Groups ◽

Abelian Groups ◽

Nilpotent Groups ◽

Generating Set ◽

Odd Order ◽

Impartial Game ◽

A Finite Group

AbstractWe study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form{T\times H}, whereTis a 2-group andHis a group of odd order. This includes all nilpotent and hence abelian groups.

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A Note on Special Local 2-Nilpotent Groups and the Solvability of Finite Groups

Algebra Colloquium ◽

10.1142/s1005386718000378 ◽

2018 ◽

Vol 25 (04) ◽

pp. 541-546

Author(s):

Jiangtao Shi ◽

Klavdija Kutnar ◽

Cui Zhang

Keyword(s):

Finite Group ◽

Nilpotent Group ◽

Finite Groups ◽

Nilpotent Groups ◽

Quaternion Group ◽

A Finite Group

A finite group G is called a special local 2-nilpotent group if G is not 2-nilpotent, the Sylow 2-subgroup P of G has a section isomorphic to the quaternion group of order 8, [Formula: see text] and NG(P) is 2-nilpotent. In this paper, it is shown that SL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if [Formula: see text], and that GL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if q is odd. Moreover, the solvability of finite groups is also investigated by giving two generalizations of a result from [A note on p-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555–1560].

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On residuals of finite groups

Journal of Group Theory ◽

10.1515/jgth-2020-0077 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Stefanos Aivazidis ◽

Thomas Müller

Keyword(s):

Finite Group ◽

Finite Groups ◽

Nilpotent Groups ◽

Simple Groups ◽

Fitting Subgroup ◽

Soluble Groups ◽

A Finite Group ◽

Fitting Formation ◽

Original Theorem

Abstract Theorem C in [S. Dolfi, M. Herzog, G. Kaplan and A. Lev, The size of the solvable residual in finite groups, Groups Geom. Dyn. 1 (2007), 4, 401–407] asserts that, in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order and that the inequality is sharp. Inspired by this result and some of the arguments in the above article, we establish the following generalisation: if 𝔛 is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and X ¯ \overline{\mathfrak{X}} is the extension-closure of 𝔛, then there exists an (explicitly known and optimal) constant 𝛾 depending only on 𝔛 such that, for all non-trivial finite groups 𝐺 with trivial 𝔛-radical, | G X ¯ | > | G | γ \lvert G^{\overline{\mathfrak{X}}}\rvert>\lvert G\rvert^{\gamma} , where G X ¯ G^{\overline{\mathfrak{X}}} is the X ¯ \overline{\mathfrak{X}} -residual of 𝐺. When X = N \mathfrak{X}=\mathfrak{N} , the class of finite nilpotent groups, it follows that X ¯ = S \overline{\mathfrak{X}}=\mathfrak{S} , the class of finite soluble groups; thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson’s classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations 𝔛 of full characteristic such that S ⊂ X ¯ ⊂ E \mathfrak{S}\subset\overline{\mathfrak{X}}\subset\mathfrak{E} , where 𝔈 denotes the class of all finite groups, thus providing applications of our main result beyond the reach of the above theorem.

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Nilpotent injectors in finite groups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700009965 ◽

1985 ◽

Vol 32 (2) ◽

pp. 293-297 ◽

Cited By ~ 6

Author(s):

Peter Förster

Keyword(s):

Finite Group ◽

Conjugacy Class ◽

Maximal Subgroup ◽

Finite Groups ◽

Nilpotent Groups ◽

Fitting Class ◽

Basic Properties ◽

A Finite Group ◽

Class F

Nilpotent injectors exist in all finite groups.For every Fitting class F of finite groups (see [2]), InjF(G) denotes the set of all H ≤ G such that for each N ⊴ ⊴ G , H ∩ N is an F -maximal subgroup of N (that is, belongs to F and i s maximal among the subgroups of N with this property). Let W and N* denote the Fitting class of all nilpotent and quasi-nilpotent groups, respectively. (For the basic properties of quasi-nilpotent groups, and of the N*-radical F*(G) of a finite group G3 the reader is referred to [5].,X. %13; we shall use these properties without further reference.) Blessenohl and H. Laue have shown in CJ] that for every finite group G, InjN*(G) = {H ≤ G | H ≥ F*(G) N*-maximal in G} is a non-empty conjugacy class of subgroups of G. More recently, Iranzo and Perez-Monasor have verified InjN(G) ≠ Φ for all finite groups G satisfying G = CG(E(G))E(G) (see [6]), and have extended this result to a somewhat larger class M of finite groups C(see [7]). One checks, however, that M does not contain all finite groups; for example, S5 ε M.

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Generalized Frattini Subgroups of Finite Groups. II

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-046-3 ◽

1969 ◽

Vol 21 ◽

pp. 418-429 ◽

Cited By ~ 2

Author(s):

James C. Beidleman

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Finite Groups ◽

Proper Subgroup ◽

Subnormal Subgroup ◽

Normal Subgroups ◽

Frattini Subgroup ◽

A Finite Group ◽

Proper Normal Subgroup ◽

Equivalent Conditions

The theory of generalized Frattini subgroups of a finite group is continued in this paper. Several equivalent conditions are given for a proper normal subgroup H of a finite group G to be a generalized Frattini subgroup of G. One such condition on H is that K is nilpotent for each normal subgroup K of G such that K/H is nilpotent. From this result, it follows that the weakly hyper-central normal subgroups of a finite non-nilpotent group G are generalized Frattini subgroups of G.Let H be a generalized Frattini subgroup of G and let K be a subnormal subgroup of G which properly contains H. Then H is a generalized Frattini subgroup of K.Let ϕ(G) be the Frattini subgroup of G. Suppose that G/ϕ(G) is nonnilpotent, but every proper subgroup of G/ϕ(G) is nilpotent. Then ϕ(G) is the unique maximal generalized Frattini subgroup of G.

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On semi-σ-nilpotent finite groups

Journal of Algebra and Its Applications ◽

10.1142/s0219498819502001 ◽

2019 ◽

Vol 18 (10) ◽

pp. 1950200

Author(s):

Chi Zhang ◽

Alexander N. Skiba

Keyword(s):

Finite Group ◽

Semidirect Product ◽

Finite Groups ◽

Nilpotent Groups ◽

Product Formula ◽

Nilpotent Subgroup ◽

Chief Factor ◽

Group A ◽

A Finite Group

Let [Formula: see text] be a partition of the set [Formula: see text] of all primes and [Formula: see text] a finite group. A chief factor [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-central if the semidirect product [Formula: see text] is a [Formula: see text]-group for some [Formula: see text]. [Formula: see text] is called [Formula: see text]-nilpotent if every chief factor of [Formula: see text] is [Formula: see text]-central. We say that [Formula: see text] is semi-[Formula: see text]-nilpotent (respectively, weakly semi-[Formula: see text]-nilpotent) if the normalizer [Formula: see text] of every non-normal (respectively, every non-subnormal) [Formula: see text]-nilpotent subgroup [Formula: see text] of [Formula: see text] is [Formula: see text]-nilpotent. In this paper we determine the structure of finite semi-[Formula: see text]-nilpotent and weakly semi-[Formula: see text]-nilpotent groups.

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Finite groups which are the product of two nilpotent subgroups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700043161 ◽

1973 ◽

Vol 9 (2) ◽

pp. 267-274 ◽

Cited By ~ 12

Author(s):

Fletcher Gross

Keyword(s):

Finite Group ◽

Finite Groups ◽

Upper Bound ◽

Frattini Subgroup ◽

Derived Length ◽

A Finite Group

Suppose G = AB where G is a finite group and A and B are nilpotent subgroups. It is proved that the derived length of G modulo its Frattini subgroup is at most the sum of the classes of A and B. An upper bound for the derived length of G in terms of the derived lengths of A and B also is obtained.

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A generalized Frattini subgroup

Asian-European Journal of Mathematics ◽

10.1142/s1793557121500261 ◽

2020 ◽

pp. 2150026

Author(s):

Ruslan V. Borodich

Keyword(s):

Finite Group ◽

Finite Groups ◽

Subnormal Subgroup ◽

Maximal Subgroups ◽

General Question ◽

Frattini Subgroup ◽

Local Formation ◽

Glasgow Math ◽

A Finite Group ◽

Subnormal Subgroups

In the work of Beidleman and Smith [On Frattini-like subgroups, Glasgow Math. J. 35 (1993) 95–98], the following question was raised: “If [Formula: see text] is a subnormal subgroup of a finite group [Formula: see text] containing [Formula: see text], then whether the supersolvability of [Formula: see text] follows the supersolvability of [Formula: see text]”. This problem was considered in works of Selkin [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)], Skiba [On the intersection of all maximal [Formula: see text]-subgroups of a finite group, Prob. Phys. Math. Tech. 3(4) (2010) 56–62], Ballester-Bolinches [On [Formula: see text]-subnormal subgroups and Frattini-like subgroups of a finite group, Glasgow Math. J. 36 (1994) 241–247] and many other authors (see monograph [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)]). In this paper, we give the answer to the more general question: “Let [Formula: see text] be a local formation. If [Formula: see text] is a subnormal subgroup of a group [Formula: see text], then in what case [Formula: see text] will follow from [Formula: see text]”.

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A Frattini-like subgroup

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100051057 ◽

1975 ◽

Vol 77 (2) ◽

pp. 247-257 ◽

Cited By ~ 10

Author(s):

M. J. Tomkinson

Keyword(s):

Finite Group ◽

Finite Groups ◽

Maximal Subgroups ◽

Frattini Subgroup ◽

A Finite Group

The Frattini subgroup φ(G) of a group G is the intersection of G and all its maximal subgroups. The following results for finite groups are well known:THEOREM A0. If G is a finite group, then the following three conditions are equivalent:(i) G is nilpotent,(ii) G/φ(G) is nilpotent,(iii) φ(G) ≥ G′.

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