scholarly journals Intense Automorphisms of Finite Groups

2021 ◽  
Vol 273 (1341) ◽  
Author(s):  
Mima Stanojkovski
Keyword(s):  
Prime Number ◽  
Semidirect Product ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Nilpotency Class ◽  
Content Type ◽  
Formula Group ◽  
Special Case

Let G G be a group. An automorphism of G G is called intense if it sends each subgroup of G G to a conjugate; the collection of such automorphisms is denoted by Int ⁡ ( G ) \operatorname {Int}(G) . In the special case in which p p is a prime number and G G is a finite p p -group, one can show that Int ⁡ ( G ) \operatorname {Int}(G) is the semidirect product of a normal p p -Sylow and a cyclic subgroup of order dividing p − 1 p-1 . In this paper we classify the finite p p -groups whose groups of intense automorphisms are not themselves p p -groups. It emerges from our investigation that the structure of such groups is almost completely determined by their nilpotency class: for p > 3 p>3 , they share a quotient, growing with their class, with a uniquely determined infinite 2-generated pro- p p group.

scholarly journals Modular Representations of Finite Groups with Unsaturated Split (B,N)-Pairs

1980 ◽  
Vol 32 (3) ◽  
pp. 714-733 ◽  
Author(s):  
N. B. Tinberg
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Prime Number ◽  
Semidirect Product ◽  
Finite Groups ◽  
Modular Representations ◽  
Characteristic P ◽  
Representations Of Finite Groups ◽  
A Finite Group

1. Introduction.Let p be a prime number. A finite group G = (G, B, N, R, U) is called a split(B, N)-pair of characteristic p and rank n if(i) G has a (B, N)-pair (see [3, Definition 2.1, p. B-8]) where H= B ⋂ N and the Weyl group W= N/H is generated by the set R= ﹛ω 1,… , ω n) of “special generators.”(ii) H= ⋂n∈N n-1Bn(iii) There exists a p-subgroup U of G such that B = UH is a semidirect product, and H is abelian with order prime to p.A (B, N)-pair satisfying (ii) is called a saturated (B, N)-pair. We call a finite group G which satisfies (i) and (iii) an unsaturated split (B, N)- pair. (Unsaturated means “not necessarily saturated”.)

Finite groups in which every cyclic subgroup is self-normalizing in its subnormal closure

2019 ◽  
Vol 22 (5) ◽  
pp. 941-951
Author(s):  
Guohua Qian
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Formula Group ◽  
A Finite Group ◽  
Closed Formation

Abstract For a given prime p, a finite group G is said to be a {\widetilde{\mathcal{C}}_{p}} -group if every cyclic p-subgroup of G is self-normalizing in its subnormal closure. In this paper, we get some descriptions of {\widetilde{\mathcal{C}}_{p}} -groups, show that the class of {\widetilde{\mathcal{C}}_{p}} -groups is a subgroup-closed formation and that {O^{p^{\prime}}(G)} is a solvable p-nilpotent group for every {\widetilde{\mathcal{C}}_{p}} -group G. We also prove that if a finite group G is a {\widetilde{\mathcal{C}}_{p}} -group for all primes p, then every subgroup of G is self-normalizing in its subnormal closure.

scholarly journals Odd order nilpotent groups of class two with cyclic centre

1974 ◽  
Vol 17 (2) ◽  
pp. 142-153 ◽  
Author(s):  
Y. K. Leong
Keyword(s):  
Nilpotent Group ◽  
Direct Product ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Nilpotency Class ◽  
Central Product ◽  
Odd Order ◽  
Image Position

The isomorphism problem for finite groups of odd order and nilpotency class 2 with cyclic centre will be solved using some results of Brady [1], [2]. Since a finite nilpotent group is the direct product of its Sylow subgroups, we only need to consider finite q-groups where q is a prime. It has been shown in [1] and [2] that a finite q-group of nilpotency class 2 with cyclic centre is a central product either of two-generator subgroups with cyclic centre or of two-generator subgroups with cyclic centre and a cyclic subgroup, and that the q-groups of class 2 on two generators with cyclic centre comprise the following list: , and if q = 2 we have as well .

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiuya Wang
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Abelian Groups ◽  
Class Groups ◽  
Abelian Extensions ◽  
Pointwise Bound ◽  
First Case

AbstractElementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

scholarly journals Polygonal products of polycyclic by finite groups

1996 ◽  
Vol 54 (3) ◽  
pp. 369-372 ◽  
Author(s):  
R.B.J.T. Allenby
Keyword(s):  
Finite Groups ◽  
Cyclic Subgroup ◽  
Substantial Improvement ◽  
Normal Subgroups ◽  
Recent Theorem

We prove that a polygonal product of polycyclic by finite groups amalgamating normal subgroups, with trivial mutual intersections, is cyclic subgroup separable. Because of a recent example (stated below) of the author this substantial improvement on a recent theorem of Kim is essentially best possible.

n-RECOGNITION BY PRIME GRAPH OF THE SIMPLE GROUP PSL(2,q)

2008 ◽  
Vol 07 (06) ◽  
pp. 735-748 ◽  
Author(s):  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Prime Number ◽  
Finite Groups ◽  
Prime Graph ◽  
Split Extension ◽  
A Finite Group

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. It is proved that if p > 11 and p ≢ 1 (mod 12), then PSL(2,p) is uniquely determined by its prime graph. Also it is proved that if p > 7 is a prime number and Γ(G) = Γ(PSL(2,p2)), then G ≅ PSL(2,p2) or G ≅ PSL(2,p2).2, the non-split extension of PSL(2,p2) by ℤ2. In this paper as the main result we determine finite groups G such that Γ(G) = Γ(PSL(2,q)), where q = pk. As a consequence of our results we prove that if q = pk, k > 1 is odd and p is an odd prime number, then PSL(2,q) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.

FINITE GROUPS WITH GIVEN NEARLY S-QUASINORMAL SUBGROUPS

2008 ◽  
Vol 01 (03) ◽  
pp. 369-382
Author(s):  
Nataliya V. Hutsko ◽  
Vladimir O. Lukyanenko ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Saturated Formation ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
Quasinormal Subgroup ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ HsG. Our main result here is the following theorem. Let [Formula: see text] be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that [Formula: see text]. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then [Formula: see text].

scholarly journals Group Laws Implying Virtual Nilpotence

2003 ◽  
Vol 74 (3) ◽  
pp. 295-312 ◽  
Author(s):  
R. G. Burns ◽  
Yuri Medvedev
Keyword(s):  
Large Class ◽  
Finite Groups ◽  
Additional Condition ◽  
Metabelian Group ◽  
Nilpotency Class ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Locally Graded Groups ◽  
Finite Exponent ◽  
Group Law

AbstractIf ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

scholarly journals Cohomology of p-groups of nilpotency class smaller than p

2018 ◽  
Vol 21 (2) ◽  
pp. 337-350 ◽  
Author(s):  
Antonio Díaz Ramos ◽  
Oihana Garaialde Ocaña ◽  
Jon González-Sánchez
Keyword(s):  
Prime Number ◽  
Nilpotency Class ◽  
Cohomology Algebra

AbstractLetpbe a prime number, letdbe an integer and letGbe ad-generated finitep-group of nilpotency class smaller thanp. Then the number of possible isomorphism types for the modpcohomology algebra{H^{*}(G;{\mathbb{F}}_{p})}is bounded in terms ofpandd.

On Pólya's Theorem

1967 ◽  
Vol 19 ◽  
pp. 792-799 ◽  
Author(s):  
J. Sheehan
Keyword(s):  
Finite Groups ◽  
Scalar Product ◽  
Combinatorial Analysis ◽  
Simple Extension ◽  
Group Characters ◽  
Polya's Theorem ◽  
Representations Of Finite Groups ◽  
Pólya’S Theorem ◽  
Special Case ◽  
Superposition Theorem

In 1927 J. H. Redfield (9) stressed the intimate interrelationship between the theory of finite groups and combinatorial analysis. With this in mind we consider Pólya's theorem (7) and the Redfield-Read superposition theorem (8, 9) in the context of the theory of permutation representations of finite groups. We show in particular how the Redfield-Read superposition theorem can be deduced as a special case from a simple extension of Pólya's theorem. We give also a generalization of the superposition theorem expressed as the multiple scalar product of certain group characters. In a later paper we shall give some applications of this generalization.

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