On a problem about cyclic subgroups of finite groups

Let $C(G)$ be the poset of cyclic subgroups of a finite group $G$ and let $\mathscr{P}$ be the class of $p$ -groups of order $p^{n}$ ( $n\geq 3$ ). Consider the function $\unicode[STIX]{x1D6FC}:\mathscr{P}\longrightarrow (0,1]$ given by $\unicode[STIX]{x1D6FC}(G)=|C(G)|/|G|$ . In this paper, we determine the second minimum value of $\unicode[STIX]{x1D6FC}$ , as well as the corresponding minimum points. Since the problem of finding the second maximum value of $\unicode[STIX]{x1D6FC}$ has been solved for $p=2$ , we focus on the case of odd primes in determining the second maximum.

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On cyclic subgroups of finite groups

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500004065 ◽

1982 ◽

Vol 25 (1) ◽

pp. 19-20 ◽

Cited By ~ 3

Author(s):

U. Dempwolff ◽

S. K. Wong

Keyword(s):

Finite Group ◽

Finite Groups ◽

Simple Proof ◽

Cyclic Subgroup ◽

Finite Subgroup ◽

Fitting Subgroup ◽

Cyclic Subgroups ◽

Nontrivial Subgroup ◽

A Finite Group

In [3] Laffey has shown that if Z is a cyclic subgroup of a finite subgroup G, then either a nontrivial subgroup of Z is normal in the Fitting subgroup F(G) or there exists a g in G such that Zg∩Z = 1. In this note we offer a simple proof of the following generalisation of that result:Theorem. Let G be a finite group and X and Y cyclic subgroups of G. Then there exists a g in G such that Xg∩Y⊴F(G).

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Disjoint conjugates of cyclic subgroups of finite groups

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500026328 ◽

1977 ◽

Vol 20 (3) ◽

pp. 229-232 ◽

Cited By ~ 3

Author(s):

Thomas J. Laffey

Keyword(s):

Finite Group ◽

Finite Groups ◽

Cyclic Subgroup ◽

Affirmative Answer ◽

Fitting Subgroup ◽

Cyclic Subgroups ◽

Result Theorem ◽

Nontrivial Subgroup ◽

A Finite Group

In an earlier paper (2) we considered the following question “If S is a cyclic subgroup of a finite group G and S ∩ F(G) = 1, where F(G) is the Fitting subgroup of G, does there necessarily exist a conjugate Sx of S in G with S ∩ Sx = l?” and we gave an affirmative answer for G simple or soluble. In this paper we answer the question affirmatively in general (in fact we prove a somewhat stronger result (Theorem 3)). We give an example of a group G with a cyclic subgroup S such that (i) no nontrivial subgroup of S is normal in G and (ii) no x exists for which S ∩ Sx = 1.

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A problem on cyclic subgroups of finite groups

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500010014 ◽

1973 ◽

Vol 18 (4) ◽

pp. 247-249 ◽

Cited By ~ 3

Author(s):

Thomas J. Laffey

Keyword(s):

Finite Group ◽

Finite Groups ◽

Cyclic Subgroups ◽

Image Position ◽

A Finite Group

Let G be a finite group and let S be a subgroup of G with

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FINITE GROUPS ALL OF WHOSE SECOND MAXIMAL SUBGROUPS ARE ${\mathcal{H^*}}$-GROUPS

International Journal of Mathematics ◽

10.1142/s0129167x13500274 ◽

2013 ◽

Vol 24 (04) ◽

pp. 1350027

Author(s):

XIANGGUI ZHONG

Keyword(s):

Finite Group ◽

Finite Groups ◽

Maximal Subgroups ◽

Cyclic Subgroups ◽

Group A ◽

A Finite Group

Let G be a finite group. A subgroup H of G is called weakly normal in G if Hg ≤ NG(H) implies g ∈ NG(H) for all g ∈ G. A finite group G is called an [Formula: see text]-group if all cyclic subgroups of G of order prime or 4 are weakly normal in G. In this paper, the structure of finite groups all of whose second maximal subgroups satisfy [Formula: see text]-property has been characterized.

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Finite groups in which every non-abelian subgroup is a TI-subgroup or a subnormal subgroup

Journal of Algebra and Its Applications ◽

10.1142/s0219498819501597 ◽

2019 ◽

Vol 18 (08) ◽

pp. 1950159

Author(s):

Jiangtao Shi

Keyword(s):

Finite Group ◽

Finite Groups ◽

Abelian Subgroup ◽

Subnormal Subgroup ◽

Cyclic Subgroups ◽

Abelian Subgroups ◽

A Finite Group

It is known that a TI-subgroup of a finite group may not be a subnormal subgroup and a subnormal subgroup of a finite group may also not be a TI-subgroup. For the non-abelian subgroups, we prove that if every non-abelian subgroup of a finite group [Formula: see text] is a TI-subgroup or a subnormal subgroup, then every non-abelian subgroup of [Formula: see text] must be subnormal in [Formula: see text]. We also show that the non-cyclic subgroups have the same property.

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Permutability degrees of finite groups

Filomat ◽

10.2298/fil1608165o ◽

2016 ◽

Vol 30 (8) ◽

pp. 2165-2175

Author(s):

D.E. Otera ◽

F.G. Russo

Keyword(s):

Finite Group ◽

Finite Groups ◽

Subgroup Lattice ◽

Cyclic Subgroups ◽

A Finite Group

Given a finite group G, we introduce the permutability degree of G, as pd(G) = 1/|G| |L(G)| ?X?L(G)|PG(X)|, where L(G) is the subgroup lattice of G and PG(X) the permutizer of the subgroup X in G, that is, the subgroup generated by all cyclic subgroups of G that permute with X ? L(G). The number pd(G) allows us to find some structural restrictions on G. Successively, we investigate the relations between pd(G), the probability of commuting subgroups sd(G) of G and the probability of commuting elements d(G) of G. Proving some inequalities between pd(G), sd(G) and d(G), we correlate these notions.

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FINITE GROUPS WITH SOME NON-CYCLIC SUBGROUPS HAVING SMALL INDICES IN THEIR NORMALIZERS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813501417 ◽

2014 ◽

Vol 13 (04) ◽

pp. 1350141

Author(s):

KLAVDIJA KUTNAR ◽

DRAGAN MARUŠIČ ◽

JIANGTAO SHI ◽

CUI ZHANG

Keyword(s):

Finite Group ◽

Finite Groups ◽

Prime Power ◽

Cyclic Subgroup ◽

Prime Power Order ◽

Cyclic Subgroups ◽

A Finite Group

In this paper, it is shown that a finite group G is always supersolvable if |NG(H) : H| ≤ 2 for every non-cyclic subgroup H of G of prime-power order. Also, finite groups with all supersolvable non-cyclic subgroups being self-normalizing, and finite p-groups with all non-cyclic proper subgroups being of prime index in their normalizers are completely classified.

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Finite groups with some weakly pronormal subgroups

Open Mathematics ◽

10.1515/math-2020-0125 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1742-1747

Author(s):

Jianjun Liu ◽

Mengling Jiang ◽

Guiyun Chen

Keyword(s):

Finite Group ◽

Finite Groups ◽

A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

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A on simplicity criterion for finite groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700007606 ◽

1969 ◽

Vol 10 (3-4) ◽

pp. 359-362

Author(s):

Nita Bryce

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Finite Groups ◽

Odd Order ◽

A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

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