scholarly journals Two remarks on Wall's D2 problem

2018 ◽  
Vol 167 (02) ◽  
pp. 361-368
Author(s):  
IAN HAMBLETON
Keyword(s):  
Finite Group ◽  
Homotopy Type ◽  
Homotopy Equivalent ◽  
Finite Complex ◽  
Simple Homotopy ◽  
A Finite Group

AbstractIf a finite group G is isomorphic to a subgroup of SO(3), then G has the D2-property. Let X be a finite complex satisfying Wall's D2-conditions. If π1(X) = G is finite, and χ(X) ≥ 1 − def(G), then X ∨ S2 is simple homotopy equivalent to a finite 2-complex, whose simple homotopy type depends only on G and χ(X).

scholarly journals Some remarks on the p-homotopy type of B∑p2

1996 ◽  
Vol 38 (3) ◽  
pp. 337-342
Author(s):  
Maurizio Brunetti
Keyword(s):  
Finite Group ◽  
Homotopy Type ◽  
Sylow Subgroup ◽  
Prime Power ◽  
Homotopy Equivalent ◽  
Homotopy Equivalence ◽  
Image Position ◽  
A Finite Group

Let G be a finite group, H a copy of its p-Sylow subgroup, and N the normalizer of H in G. A theorem by Nishida [10] states the p-homotopy equivalence of suitable suspensions of BN and BG when H is abelian. Recently, in [3] the authors proved a stronger result: let ΩkH be the subgroup of H generated by elements of order pk or less; ifthen BN and BG are stably p-homotopy equivalent. The hypothesis above is obviously verified when H is abelian. In the same paper the authors recall that H does not verify such condition when p = 2 and G = SL2(Fq) for a suitable odd prime power q; in this case BG and BN are not stably 2-homotopy equivalent.

The centric p-radical complex and a related p-local geometry

2002 ◽  
Vol 133 (3) ◽  
pp. 383-398 ◽  
Author(s):  
MASATO SAWABE
Keyword(s):  
Finite Group ◽  
Prime Divisor ◽  
Homotopy Equivalent ◽  
Local Geometry ◽  
Group Of Lie Type ◽  
Radical Complex ◽  
A Finite Group ◽  
Natural Hypothesis

We shall introduce a p-local geometry Δp(G) for the pair (G, p) of a finite group G and a prime divisor p of the order of G, which is constructed by ‘maximal parabolic like’ subgroups of G. Under the natural hypothesis, Δp(G) behaves very much like the building associated with a group of Lie type in characteristic p. We shall also show that, under some hypothesis, Δp(G) is homotopy equivalent to the subgroup complex [Bscr ]cenp(G) which is a more essential part of the p-radical complex [Bscr ]p(G). Some of the p-local sporadic geometries can be understood well in our system.

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

scholarly journals Finite groups with some weakly pronormal subgroups

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.

scholarly journals Random additions in urns of integers

2021 ◽  
Vol 58 (2) ◽  
pp. 335-346
Author(s):  
Mackenzie Simper
Keyword(s):  
Finite Group ◽  
Random Process ◽  
Uniform Distribution ◽  
Discrete Time ◽  
Random Variable ◽  
Urn Model ◽  
Infinite Type ◽  
The Mean ◽  
A Finite Group ◽  
Proof Of Convergence

AbstractConsider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

𝒫-characters and the structure of finite solvable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiakuan Lu ◽  
Kaisun Wu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Irreducible Constituent ◽  
A Finite Group

AbstractLet 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

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