scholarly journals High rank torus actions on contact manifolds

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Gianluca Occhetta ◽  
Eleonora A. Romano ◽  
Luis E. Solá Conde ◽  
Jarosław A. Wiśniewski
Keyword(s):  
Simple Group ◽  
High Rank ◽  
Linear Groups ◽  
Fano Manifold ◽  
Contact Manifolds ◽  
Torus Actions ◽  
Group Of Automorphisms ◽  
Almost All

AbstractWe prove LeBrun–Salamon conjecture in the following situation: if X is a contact Fano manifold of dimension $$2n+1$$ 2 n + 1 whose group of automorphisms is reductive of rank $$\ge \max (2,(n-3)/2)$$ ≥ max ( 2 , ( n - 3 ) / 2 ) then X is the adjoint variety of a simple group. The rank assumption is fulfilled not only by the three series of classical linear groups but also by almost all the exceptional ones.

ON SPECTRUM OF LINEAR GROUPS OVER THE BINARY FIELD AND RECOGNIZABILITY OF L12(2)

2006 ◽  
Vol 16 (02) ◽  
pp. 341-349 ◽  
Author(s):  
A. R. MOGHADDAMFAR
Keyword(s):  
Finite Group ◽  
Computer Program ◽  
Simple Group ◽  
Linear Groups ◽  
Element Orders ◽  
Binary Field ◽  
Set Of Element Orders ◽  
A Finite Group

The spectrum ω(G) of a finite group G is the set of element orders of G. A finite group G is said to be recognizable through its spectrum, if for every finite group H, the equality of the spectra ω(H) = ω(G) implies the isomorphism H ≅ G. In this paper, first we try to write a computer program for computing ω(Ln(2)) for any n ≥ 3. Then, we will show that the simple group L12(2) is recognizable through its spectrum.

A Conjecture of Bachmuth and Mochizuki on Automorphisms of Soluble Groups

1976 ◽  
Vol 28 (6) ◽  
pp. 1302-1310 ◽  
Author(s):  
Brian Hartley
Keyword(s):  
Finite Index ◽  
Soluble Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Linear Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Finitely Generated Group ◽  
Soluble Subgroup

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.

scholarly journals On isomorphisms of connected Cayley graphs, III

1998 ◽  
Vol 58 (1) ◽  
pp. 137-145 ◽  
Author(s):  
Cai Heng Li
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Simple Group ◽  
Prime Divisor ◽  
Cayley Graphs ◽  
Simple Groups ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Natural Question ◽  
A Finite Group

For a finite group G and a subset S of G which does not contain the identity of G, we use Cay(G, S) to denote the Cayley graph of G with respect to S. For a positive integer m, the group G is called a (connected) m-DCI-group if for any (connected) Cayley graphs Cay(G, S) and Cay(G, T) of out-valency at most m, Sσ = T for some σ ∈ Aut(G) whenever Cay(G, S) ≅ Cay(G, T). Let p(G) be the smallest prime divisor of |G|. It was previously shown that each finite group G is a connected m-DCI-group for m ≤ p(G) − 1 but this is not necessarily true for m = p(G). This leads to a natural question: which groups G are connected p(G)-DCI-groups? Here we conjecture that the answer of this question is positive for finite simple groups, that is, finite simple groups are all connected 2-DCI-groups. We verify this conjecture for the linear groups PSL(2, q). Then we prove that a nonabelian simple group G is a 2-DCI-group if and only if G = A5.

A NEW CHARACTERIZATION OF U4(7) BY ITS NONCOMMUTING GRAPH

2009 ◽  
Vol 08 (01) ◽  
pp. 105-114 ◽  
Author(s):  
LIANGCAI ZHANG ◽  
WUJIE SHI
Keyword(s):  
Simple Group ◽  
Simple Groups ◽  
Nonabelian Simple Group ◽  
Prime Graphs ◽  
Vertex Set ◽  
Thompson’S Conjecture ◽  
A Finite Group ◽  
Noncommuting Graph ◽  
Almost All

Let G be a finite nonabelian group and associate a disoriented noncommuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, J. G. Thompson gave the following conjecture.Thompson's Conjecture If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G) denotes the set of the sizes of the conjugacy classes of G.In 2006, A. Abdollahi, S. Akbari and H. R. Maimani put forward a conjecture in [1] as follows.AAM's Conjecture Let M be a finite nonabelian simple group and G a group such that ∇(G)≅ ∇ (M). Then G ≅ M.Even though both of the two conjectures are known to be true for all finite simple groups with nonconnected prime graphs, it is still unknown for almost all simple groups with connected prime graphs. In the present paper, we prove that the second conjecture is true for the projective special unitary simple group U4(7).

ON ISOMORPHISMS OF VERTEX-TRANSITIVE CUBIC GRAPHS

2015 ◽  
Vol 99 (3) ◽  
pp. 341-349
Author(s):  
JING CHEN ◽  
CAI HENG LI ◽  
WEI JUN LIU
Keyword(s):  
Simple Group ◽  
Isomorphism Problem ◽  
Cubic Graphs ◽  
Group Of Automorphisms ◽  
Vertex Transitive

We study the isomorphism problem of vertex-transitive cubic graphs which have a transitive simple group of automorphisms.

scholarly journals On the uniform spread of almost simple linear groups

2013 ◽  
Vol 209 ◽  
pp. 35-109 ◽  
Author(s):  
Timothy C. Burness ◽  
Simon Guest
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Simple Group ◽  
Nonnegative Integer ◽  
Finite Simple Group ◽  
Linear Groups ◽  
A Finite Group

AbstractLet G be a finite group, and let k be a nonnegative integer. We say that G has uniform spread k if there exists a fixed conjugacy class C in G with the property that for any k nontrivial elements x1,…,xk in G there exists y ∊ C such that G = ‹xi,y› for all i. Further, the exact uniform spread of G, denoted by u(G), is the largest k such that G has the uniform spread k property. By a theorem of Breuer, Guralnick, and Kantor, u(G) ≥ 2 for every finite simple group G. Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if G = ‹PSLn (q),g› is almost simple, then u(G) ≥ 2 (unless G ≅ S6), and we determine precisely when u(G) tends to infinity as |G| tends to infinity.

On the linearity of the group of automorphisms of linear groups

1971 ◽  
Vol 10 (5) ◽  
pp. 313-324
Author(s):  
Yu. I. Merzlyakov
Keyword(s):  
Linear Groups ◽  
Group Of Automorphisms

scholarly journals On the Codes Related to the Higman-Sims Graph

10.37236/4267 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Wolfgang Knapp ◽  
Hans-Jörg Schaeffer
Keyword(s):  
Simple Group ◽  
Linear Codes ◽  
Maximal Subgroups ◽  
Dual Codes ◽  
Golay Code ◽  
Weight Enumerators ◽  
Low Weight ◽  
Vertex Set ◽  
Almost All ◽  
Sims Graph

All linear codes of length $100$ over a field $F$ which admit the Higman-Sims simple group HS in its rank $3$ representation are determined. By group representation theory it is proved that they can all be understood as submodules of the permutation module $F\Omega$ where $\Omega$ denotes the vertex set of the Higman-Sims graph. This module is semisimple if $\mathrm{char} F\neq 2,5$ and absolutely indecomposable otherwise. Also if $\mathrm{char} F \in \{2, 5\}$ the submodule lattice is determined explicitly. The binary case $F = \mathbb{F}_2$ is studied in detail under coding theoretic aspects. The HS-orbits in the subcodes of dimension $\leq 23$ are computed explicitly and so also the weight enumerators are obtained. The weight enumerators of the dual codes are determined by MacWilliams transformation. Two fundamental methods are used: Let $v$ be the endomorphism determined by an adjacency matrix. Then in $H_{22} = \mathrm{Im} v $ the HS-orbits are determined as $v$-images of certain low weight vectors in $F\Omega$ which carry some special graph configurations. The second method consists in using the fact that $H_{23}/H_{21}$ is a Klein four group under addition, if $H_{23}$ denotes the code generated by $H_{22}$ and a "Higman vector" $x(m)$ of weight 50 associated to a heptad $m$ in the shortened Golay code $G_{22}$, and $H_{21}$ denotes the doubly even subcode of $H_{22}\leq H_{78} = {H_{22}}^\perp$. Using the mentioned observation about $H_{23}/H_{21}$ and the results on the HS-orbits in $H_{23}$ a model of G. Higman's geometry is constructed, which leads to a direct geometric proof that G. Higman's simple group is isomorphic to HS. Finally, it is shown that almost all maximal subgroups of the Higman-Sims group can be understood as stabilizers in HS of codewords in $H_{23}$.

scholarly journals A convexity theorem for torus actions on contact manifolds

2002 ◽  
Vol 46 (1) ◽  
pp. 171-184 ◽  
Author(s):  
Eugene Lerman
Keyword(s):  
Convexity Theorem ◽  
Contact Manifolds ◽  
Torus Actions

scholarly journals Random bases for coprime linear groups

2020 ◽  
Vol 23 (1) ◽  
pp. 133-157
Author(s):  
Hülya Duyan ◽  
Zoltán Halasi ◽  
Károly Podoski
Keyword(s):  
Group Theory ◽  
Permutation Group ◽  
Linear Group ◽  
Additional Assumption ◽  
Probabilistic Method ◽  
Personal Communication ◽  
Linear Groups ◽  
Minimal Base ◽  
Base Size ◽  
Almost All

AbstractThe minimal base size {b(G)} for a permutation group G is a widely studied topic in permutation group theory. Z. Halasi and K. Podoski [Every coprime linear group admits a base of size two, Trans. Amer. Math. Soc. 368 2016, 8, 5857–5887] proved that {b(G)\leq 2} for coprime linear groups. Motivated by this result and the probabilistic method used by T. Burness, M. W. Liebeck and A. Shalev, it was asked by L. Pyber [Personal communication, Bielefeld, 2017] whether or not, for coprime linear groups {G\leq GL(V)}, there exists a constant c such that the probability that a random c-tuple is a base for G tends to 1 as {\lvert V\rvert\to\infty}. While the answer to this question is negative in general, it is positive under the additional assumption that G is primitive as a linear group. In this paper, we show that almost all 11-tuples are bases for coprime primitive linear groups.

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