Finite Nonabelian Simple Group Research Articles

Surjective word maps and Burnside’s $$p^aq^b$$ p a q b theorem

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map $$(x,y) \mapsto x^Ny^N$$ is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map $$(x,y,z) \mapsto x^Ny^Nz^N$$ is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit–Thompson. We also prove asymptotic results about the surjectivity of the word map $$(x,y) \mapsto x^Ny^N$$ that depend on the number of prime factors of the integer N.

The depth of a finite simple group

We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups. We determine the simple groups of minimal depth, and show, somewhat surprisingly, that alternating groups have bounded depth. We also establish general upper bounds on the depth of simple groups of Lie type, and study the relation between the depth and the much studied notion of the length of simple groups. The proofs of our main theorems depend (among other tools) on a deep number-theoretic result, namely, Helfgott's recent solution of the ternary Goldbach conjecture.

Pentavalent symmetric graphs admitting transitive non-abelian characteristically simple groups

Let Γ be a graph and let G be a group of automorphisms of Γ. The graph Γ is called G-normal if G is normal in the automorphism group of Γ. Let T be a finite non-abelian simple group and let G=Tl with l≥1. In this paper we prove that if every connected pentavalent symmetric T-vertex-transitive graph is T-normal, then every connected pentavalent symmetric G-vertex-transitive graph is G-normal. This result, among others, implies that every connected pentavalent symmetric G-vertex-transitive graph is G-normal except T is one of 57 simple groups. Furthermore, every connected pentavalent symmetric G-regular graph is G-normal except T is one of 20 simple groups, and every connected pentavalent G-symmetric graph is G-normal except T is one of 17 simple groups.

English

Let G be a finite group. An element g ∈ G is called a vanishing element if there exists an irreducible complex character χ of G such that χ(g)= 0. Denote by Vo(G) the set of orders of vanishing elements of G. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo(G) = Vo(M) and |G| = |M|. Then G ≌ M. We answer in affirmative this conjecture for M = Sz(q), where q = 22n+1 and either q − 1, \(q - \sqrt {2q} + 1\) or q + \(\sqrt {2q} + 1\) is a prime number, and M = F4(q), where q = 2 n and either q4 + 1 or q4 − q2 + 1 is a prime number.

Exploring Infinite Groups in This Thesis Was to Discover Homomorphic Images of Non-Abelian Finite Simple Groups

To create non-abelian finite simple groups as homomorphic pictures, this thesis explores infinite groups. These semidirect products, or progenitors, are infinite groups. Researched ancestors x of x permutations: x  &nbsp

The solvable subgroups of large order of L2(p) , p≥5

By using the following theoretical and computational algorithms , we determined the solvable subgroups of large order of the finite non-abelian simple linear groups G = L2(p) = PSL(2,p) , for p≥5 and p is a prime number , also their presentations and permutation representations have been found .

Fibers of word maps and the multiplicities of non-abelian composition factors

We call a reduced word [Formula: see text] multiplicity-bounding if and only if a finite group on which the word map of [Formula: see text] has a fiber of positive proportion [Formula: see text] can only contain each non-abelian finite simple group [Formula: see text] as a composition factor with multiplicity bounded in terms of [Formula: see text] and [Formula: see text]. In this paper, based on recent work of Nikolov, we present methods to show that a given reduced word is multiplicity-bounding and apply them to give some nontrivial examples of multiplicity-bounding words, such as words of the form [Formula: see text], where [Formula: see text] is a single variable and [Formula: see text] an odd integer.

Cubic graphical regular representations of finite non-abelian simple groups

ABSTRACTGiven a finite group R, a graphical regular representation of R is a Cayley graph Γ over R with R = Aut(Γ). In this paper we study graphical regular representations of finite non-abelian simple groups of small valency.

English

For a finite group G denote by N(G) the set of conjugacy class sizes of G. In 1980s, J.G.Thompson posed the following conjecture: If L is a finite nonabelian simple group, G is a finite group with trivial center and N(G) = N(L), then G ≅ L. We prove this conjecture for an infinite class of simple groups. Let p be an odd prime. We show that every finite group G with the property Z(G) = 1 and N(G) = N(A i ) is necessarily isomorphic to A i , where i ∈ {2p, 2p + 1}.

The involution width of finite simple groups

For a finite group generated by involutions, the involution width is defined to be the minimal k∈N such that any group element can be written as a product of at most k involutions. We show that the involution width of every non-abelian finite simple group is at most 4. This result is sharp, as there are families with involution width precisely 4.

SIMPLE GROUPS, PRODUCT ACTIONS, AND GENERALIZED QUADRANGLES

The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group$G$preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on$G$, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that$G$cannot haveholomorph compoundO’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.

On Arc-Transitive Pentavalent Cayley Graphs on Finite Nonabelian Simple Groups

A Cayley graph $${\varGamma }=\mathsf{Cay}(G,S)$$ is said to be normal if G is normal in $$\mathsf{Aut}{\varGamma }$$ . The concept of normal Cayley graphs was first proposed by Xu (Discrete Math 182:309–319, 1998) and it plays an important role in determining the full automorphism groups of Cayley graphs. In this paper, we study the normality of connected arc-transitive pentavalent Cayley graphs $${\varGamma }$$ on finite nonabelian simple groups G, where the vertex stabilizer $$\mathsf{A}_v$$ is soluble for $$\mathsf{A}=\mathsf{Aut}{\varGamma }$$ and $$v\in V{\varGamma }$$ . We prove that $${\varGamma }$$ is either normal or $$G=\mathsf{A}_{39}$$ or $$\mathsf{A}_{79}$$ . Further, a connected pentavalent arc-transitive non-normal Cayley graph on $$\mathsf{A}_{79}$$ is constructed. To our knowledge, this is the first known example of pentavalent 3-arc-transitive Cayley graph on finite nonabelian simple group which is non-normal.

Some conjectures on Frobenius' character sum

Let G be a finite group and let g ∈ G . In 1896, Frobenius showed that the number of ways to express g as a commutator of elements of G is | G | F G ( g ) , where F G ( g ) = ∑ χ ∈ Irr ( G ) χ ( g ) / χ ( 1 ) is the Frobenius character sum. This sum received particular attention in the case where G is a (non-abelian) finite simple group, and some related conjectures were posed. In this paper we discuss these conjectures, refute one of them, and provide partial evidence in favor of another one.

A generalisation of a theorem of Wielandt

In 1974, Helmut Wielandt proved that in a finite group G, a subgroup A is subnormal if and only if it is subnormal in 〈A,g〉 for all g∈G. In this paper, we prove that the subnormality of an odd order nilpotent subgroup A of G is already guaranteed by a seemingly weaker condition: A is subnormal in G if for every conjugacy class C of G there exists c∈C for which A is subnormal in 〈A,c〉. We also prove the following property of finite non-abelian simple groups: if A is a subgroup of odd prime order p in a finite almost simple group G, then there exists a cyclic p′-subgroup of F⁎(G) which does not normalise any non-trivial p-subgroup of G that is generated by conjugates of A.

Fibers of automorphic word maps and an application to composition factors

Abstract In this paper, we study the fibers of “automorphic word maps”, a certain generalization of word maps, on finite groups and on nonabelian finite simple groups in particular. As an application, we derive a structural restriction on finite groups G where, for some fixed nonempty reduced word w in d variables and some fixed ρ ∈ ( 0 , 1 ] {\rho\in(0,1]} , the word map w G {w_{G}} on G has a fiber of size at least ρ ⁢ | G | d {\rho|G|^{d}} : No sufficiently large alternating group and no (classical) simple group of Lie type of sufficiently high rank can occur as a composition factor of such a group G.

On the non-nilpotent graphs of a group

Let $G$ be a group and $nil(G)=\{x \in G \mid \langle x,y \rangle \text{ is nilpotent for all }\\ y \in G\}$. Associate a graph $\mathfrak{R}_G$ (called the non-nilpotent graph of $G$) with $G$ as follows: Take $G \setminus nil(G)$ as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of $\mathfrak{R}_G$. We conjecture that the domination number of the non-nilpotent graph of every finite non-abelian simple group is 2. We also conjecture that if $G$ and $H$ are two non-nilpotent finite groups such that $\mathfrak{R}_G\cong \mathfrak{R}_H$, then $|G| = |H|$. Among other results, we show that the non-nilpotent graph of $D_{10}$ is double-toroidal.

Several quantitative characterizations of some specific groups

Let $G$ be a finite group and let $\pi(G)=\{p_1, p_2,\ldots, p_k\}$ be the set of prime divisors of $|G|$ for which $p_1< p_2< \cdots < p_k$. The Gruenberg-Kegel graph of $G$, denoted $\operatorname{GK} (G)$, is defined as follows: its vertex set is $\pi(G)$ and two different vertices $p_i$ and $p_j$ are adjacent by an edge if and only if $G$ contains an element of order $p_i p_j$. The degree of a vertex $p_i$ in ${\rm GK}(G)$ is denoted by $d_G(p_i)$ and the $k$-tuple $D(G)= (d_G(p_1), d_G(p_2),\ldots, d_G(p_k))$ is said to be the degree pattern of $G$. Moreover, if $\omega \subseteq \pi(G)$ is the vertex set of a connected component of $\operatorname{GK} (G)$, then the largest $\omega$-number which divides $|G|$, is said to be an order component of $\operatorname{GK} (G)$. We will say that the problem of OD-characterization is solved for a finite group if we find the number of pairwise non-isomorphic finite groups with the same order and degree pattern as the group under study. The purpose of this article is twofold. First, we completely solve the problem of OD-characterization for every finite non-abelian simple group with orders having prime divisors at most 29. In particular, we show that there are exactly two non-isomorphic finite groups with the same order and degree pattern as $U_4(2)$. Second, we prove that there are exactly two non-isomorphic finite groups with the same order components as $U_5(2)$.

Bounded Normal Generation for Projective Unitary Groups of Certain Infinite Operator Algebras

We study the question how quickly products of a fixed conjugacy class cover the entire group in the projective unitary group of the connected component of the identity of the Calkin algebra, as well as the projective unitary group of a factor von Neumann algebra of type III. Our result is that the number of factors that are needed is as small as permitted by the (essential) operator norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups and analogous results for unitary groups of II_1-factors.

Mixing for three-term progressions in finite simple groups

AbstractAnswering a question of Gowers, Tao proved that any A × B × C ⊂ SLd(𝔽q)3 contains |A||B||C|/|SLd(𝔽q)| + Od(|SLd(𝔽q)|2/qmin(d−1,2)/8) three-term progressions (x, xy, xy2). Using a modification of Tao's argument, we prove such a mixing result for three-term progressions in all nonabelian finite simple groups except for PSL2(𝔽q) with an error term that depends on the degree of quasirandomness of the group. This argument also gives an alternative proof of Tao's result when d &gt; 2, but with the error term O(|SLd(𝔽q)|2/q(d−1)/24).

Answer to a 1962 question by Zappa on cosets of Sylow subgroups

In a paper in 1962, Guido Zappa asked whether a non-trivial coset of a Sylow p-subgroup of a finite group could contain only elements whose orders are powers of p, and also in that case, at least one element of order p. The first question was raised again recently in a 2014 paper by Daniel Goldstein and Robert Guralnick, when generalising an answer by John Thompson in 1967 to a similar question by L.J. Paige. In this paper we give a positive answer to both questions of Zappa, showing somewhat surprisingly that in a number of non-abelian finite simple groups (including PSL(3,4), PSU(5,2) and the Janko group J3), some non-trivial coset of a Sylow 5-subgroup (of order 5) contains only elements of order 5. Also Zappa's first question is studied in more detail. Various possibilities for the group and its Sylow p-subgroup P are eliminated, and it then follows that |P|≥5 and |P|≠8. It is an open question as to whether the order of the Sylow p-subgroup can be 7 or 9 or more.