Towards a classification of rigid product quotient varieties of Kodaira dimension 0

AbstractIn this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group G. It is shown that only for $$G = {{\,\mathrm{He}\,}}(3), {\mathbb {Z}}_3^2$$ G = He ( 3 ) , Z 3 2 , and only for dimension $$\ge 4$$ ≥ 4 such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension 4 is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

On the Frobenius spectrum of a finite group

10.1142/s0219498817500517 ◽

2017 ◽

Vol 16 (03) ◽

pp. 1750051 ◽

Cited By ~ 1

Author(s):

Jiangtao Shi ◽

Wei Meng ◽

Cui Zhang

Keyword(s):

Finite Group ◽

Positive Integer ◽

Finite Groups ◽

Prime Divisor ◽

Complete Classification ◽

Composition Factor ◽

Frobenius Theorem ◽

Even Order ◽

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

Automorphism orbits of finite groups

10.1017/s1446788700027221 ◽

1986 ◽

Vol 40 (2) ◽

pp. 253-260 ◽

Cited By ~ 9

Author(s):

Thomas J. Laffey ◽

Desmond MacHale

Keyword(s):

Finite Group ◽

Automorphism Group ◽

Finite Groups ◽

Prime Power ◽

Structure Theorem ◽

Equivalence Classes ◽

Prime Power Order ◽

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

Proper connection of power graphs of finite groups

10.1142/s021949882150033x ◽

2020 ◽

pp. 2150033

Author(s):

Xuanlong Ma

Keyword(s):

Finite Group ◽

Finite Groups ◽

Undirected Graph ◽

The Other ◽

Connection Number ◽

Power Graph ◽

Vertex Set ◽

Proper Connection Number ◽

Let [Formula: see text] be a finite group. The power graph of [Formula: see text] is the undirected graph whose vertex set is [Formula: see text], and two distinct vertices are adjacent if one is a power of the other. The reduced power graph of [Formula: see text] is the subgraph of the power graph of [Formula: see text] obtained by deleting all edges [Formula: see text] with [Formula: see text], where [Formula: see text] and [Formula: see text] are two distinct elements of [Formula: see text]. In this paper, we determine the proper connection number of the reduced power graph of [Formula: see text]. As an application, we also determine the proper connection number of the power graph of [Formula: see text].

Finite automorphism groups of laminated near-rings

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500004351 ◽

1983 ◽

Vol 26 (3) ◽

pp. 297-306 ◽

Cited By ~ 2

Author(s):

K. D. Magill ◽

P. R. Misra ◽

U. B. Tewari

Keyword(s):

Finite Group ◽

Linear Group ◽

Complex Plane ◽

Finite Groups ◽

Automorphism Groups ◽

The Other ◽

Complex Polynomial ◽

Infinite Groups ◽

Nonsingular Matrices ◽

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

On the minimum degree of the power graph of a finite cyclic group

10.1142/s0219498821500444 ◽

2020 ◽

pp. 2150044

Author(s):

Ramesh Prasad Panda ◽

Kamal Lochan Patra ◽

Binod Kumar Sahoo

Keyword(s):

Finite Group ◽

Cyclic Group ◽

Finite Groups ◽

Minimum Degree ◽

Simple Graph ◽

The Other ◽

Prime Divisors ◽

Power Graph ◽

Vertex Set ◽

The power graph [Formula: see text] of a finite group [Formula: see text] is the undirected simple graph whose vertex set is [Formula: see text], in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer [Formula: see text], let [Formula: see text] denote the cyclic group of order [Formula: see text] and let [Formula: see text] be the number of distinct prime divisors of [Formula: see text]. The minimum degree [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For [Formula: see text], under certain conditions involving the prime divisors of [Formula: see text], we identify at most [Formula: see text] vertices such that [Formula: see text] is equal to the degree of at least one of these vertices. If [Formula: see text], or that [Formula: see text] is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of [Formula: see text].

CLASSIFICATION OF FINITE GROUPS VIA THEIR BREADTH

Glasgow Mathematical Journal ◽

10.1017/s0017089519000272 ◽

2019 ◽

Vol 62 (3) ◽

pp. 544-563 ◽

Cited By ~ 1

Author(s):

HERMANN HEINEKEN ◽

FRANCESCO G. RUSSO

Keyword(s):

Inverse Problem ◽

Finite Group ◽

Finite Groups ◽

AbstractLet k be a divisor of a finite group G and Lk(G) = {x ∈ G | xk =1}. Frobenius proved that the number |Lk(G)| is always divisible by k. The following inverse problem is considered: for a given integer n, find all groups G such that max{k-1|Lk(G)| | k ∈ Div(G)} = n, where Div(G) denotes the set of all divisors of |G|. A procedure beginning with (in a sense) minimal members and deducing the remaining ones is outlined and executed for n=8.

A CLASSIFICATION OF FINITE p-GROUPS WHOSE PROPER SUBGROUPS ARE OF CLASS ≤ 2 (II)

10.1142/s021949881250171x ◽

2012 ◽

Vol 12 (03) ◽

pp. 1250171 ◽

Cited By ~ 1

Author(s):

PUJIN LI

Keyword(s):

Finite Group ◽

Complete Classification ◽

A finite group G is said to be a minimal non-[Formula: see text] group if G is not a group of class ≤ n whose proper subgroups are of class ≤ n. In this paper, we give a complete classification of minimal non-[Formula: see text]p-groups G with G3 ≅ Cp × Cp for p > 3.

ON AN INVERSE PROBLEM TO FROBENIUS' THEOREM II

10.1142/s0219498812500922 ◽

2012 ◽

Vol 11 (05) ◽

pp. 1250092 ◽

Cited By ~ 4

Author(s):

WEI MENG ◽

JIANGTAO SHI ◽

KELIN CHEN

Keyword(s):

Inverse Problem ◽

Finite Group ◽

Positive Integer ◽

Upper Bound ◽

Complete Classification ◽

Frobenius Theorem ◽

Let G be a finite group and e a positive integer dividing |G|, the order of G. Denoting Le(G) = {x ∈ G|xe = 1}. Frobenius proved that |Le(G)| = ke for some positive integer k ≥ 1. Let k(G) be the upper bound of the set {k||Le(G)| = ke, ∀ e ||G|}. In this paper, a complete classification of the finite group G with k(G) = 3 is obtained.

A Class of Finite Groups with Abelian Sylow p-Subgroups

Algebra Colloquium ◽

10.1142/s1005386706000411 ◽

2006 ◽

Vol 13 (03) ◽

pp. 471-480

Author(s):

Zhikai Zhang

Keyword(s):

Finite Group ◽

Finite Groups ◽

Simple Groups ◽

Finite Simple Groups ◽

Defect Groups ◽

Abelian Defect Groups ◽

In this paper, we first determine the structure of the Sylow p-subgroup P of a finite group G containing no elements of order 2p (p > 2), and then show that the Broué Abelian Defect Groups Conjecture is true for the principal p-block of G. The result depends on the classification of finite simple groups.

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.