Non-identity Element Research Articles

The conjugacy class ranks of $M_{24}$

$M_{24}$ is the largest Mathieu sporadic simple group of order $244 823 040 = 2^{10} {cdot} 3^3 {cdot} 5 {cdot} 7 {cdot} 11 {cdot} 23$ and contains all the other Mathieu sporadic simple groups as subgroups. The object in this paper is to study the ranks of $M_{24}$ with respect to the conjugacy classes of all its nonidentity elements.

On free groups in the infinitely based varieties of S. I. Adian

We study the free groups in varieties defined by an arbitrary set of identities in a well-known infinite independent system of identities in two variables constructed by S. I. Adian to solve the finite basis problem in group theory. We prove that the centralizer of any non-identity element in a relatively free group in any of the varieties under consideration is cyclic, and for every the set of all non-isomorphic free groups of rank in these varieties is of the cardinality of the continuum. All these groups have trivial centre, all their abelian subgroups are cyclic, and all their non-trivial normal subgroups are infinite. For any free group in any of these varieties, we also obtain a description of the automorphisms of the semigroup , answering a question posed by Plotkin in 2000. In particular, we prove that the automorphism group of any such is canonically embedded in the group .

Semiregular automorphisms in vertex-transitive graphs with a solvable group of automorphisms

It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally 2-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. The known affirmative answers for graphs with primitive and quasiprimitive groups of automorphisms suggest that solvable groups need to be considered if one is to hope for a complete solution of this conjecture. It is the purpose of this paper to present an overview of known results and suggest possible further lines of research towards a complete solution of the problem.

Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block

Abstract Let G be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field F of characteristic p ≥ 0 {p\geq 0} , and let u ∈ G {u\in G} be a nonidentity unipotent element. Let ϕ be a non-trivial irreducible representation of G. Then the Jordan normal form of ϕ ⁢ ( u ) {\phi(u)} contains at most one non-trivial block if and only if G is of type G 2 {G_{2}} , u is a regular unipotent element and dim ⁡ ϕ ≤ 7 {\dim\phi\leq 7} . Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a single non-trivial block were determined by I. D. Suprunenko [21].

On the uniform spread of almost simple symplectic and orthogonal groups

A group is 32-generated if every non-identity element is contained in a generating pair. A conjecture of Breuer, Guralnick and Kantor from 2008 asserts that a finite group is 32-generated if and only if every proper quotient of the group is cyclic, and recent work of Guralnick reduces this conjecture to almost simple groups. In this paper, we prove a stronger form of the conjecture for almost simple symplectic and odd-dimensional orthogonal groups. More generally, we study the uniform spread of these groups, obtaining lower bounds and related asymptotics. This builds on earlier work of Burness and Guest, who established the conjecture for almost simple linear groups.

Residual p-finiteness of generalized free products of groups

Let p be a prime number. Recall that a group G is said to be a residually finite p-group if for every non-identity element a of G there exists a homomorphism of the group G onto a finite p-group such that the image of a does not coincide with the identity. We obtain a necessary and sufficient condition for the free product of two residually finite p-groups with finite amalgamated subgroup to be a residually finite p-group. This result is a generalization of Higman’s theorem on the free product of two finite p-groups with amalgamated subgroup.

On the influence of fixed point free nilpotent automorphism groups

A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that \( [F,h]=F\) for all nonidentity elements \(h\in H\). Let FH be a Frobenius-like group with complement H of prime order such that \(C_F(H)\) is of prime order. Suppose that FH acts on a finite group G by automorphisms where \( (|G|,|H|)=1\) in such a way that \(C_G(F)=1.\) In the present paper we prove that the Fitting series of \(C_G(H)\) coincides with the intersections of \( C_G(H)\) with the Fitting series of G, and the nilpotent length of G exceeds the nilpotent length of \(C_G(H)\) by at most one. As a corollary, we also prove that for any set of primes \(\pi \), the upper \(\pi \)-series of \( C_G(H)\) coincides with the intersections of \(C_G(H)\) with the upper \(\pi \) -series of G, and the \(\pi \)- length of G exceeds the \(\pi \)-length of \( C_G(H)\) by at most one.

Action of a Frobenius-like group with kernel having central derived subgroup

A finite group [Formula: see text] is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup [Formula: see text] with a nontrivial complement [Formula: see text] such that [Formula: see text] for all nonidentity elements [Formula: see text]. Suppose that a finite group [Formula: see text] admits a Frobenius-like group of automorphisms [Formula: see text] of coprime order with [Formula: see text] In case where [Formula: see text] we prove that the groups [Formula: see text] and [Formula: see text] have the same nilpotent length under certain additional assumptions.

Near-complete external difference families

We introduce and explore near-complete external difference families, a partitioning of the nonidentity elements of a group so that each nonidentity element is expressible as a difference of elements from distinct subsets a fixed number of times. We show that the existence of such an object implies the existence of a near-resolvable design. We provide examples and general constructions of these objects, some of which lead to new parameter families of near-resolvable designs on a non-prime-power number of points. Our constructions employ cyclotomy, partial difference sets, and Galois rings.

Finite entropy actions of free groups, rigidity of stabilizers, and a Howe-Moore type phenomenon

We study a notion of entropy, called f-invariant entropy, introduced by Lewis Bowen for probability measure preserving actions of finitely generated free groups. In the degenerate case, the f-invariant entropy is -∞. In this paper, we investigate the qualitative consequences of an action having finite f-invariant entropy. We find three main properties of such actions. First, the stabilizers occurring in factors of such actions are highly restricted. Specifically, the stabilizer of almost every point must be either trivial or of finite index. Second, such actions are very chaotic in the sense that when the space is not essentially countable, every non-identity group element acts with infinite Kolmogorov-Sinai entropy. Finally, we show that such actions display behavior reminiscent of the Howe-Moore property. Specifically, if the action is ergodic, there exists an integer n such that for every non-trivial normal subgroup K, the number of K-ergodic components is at most n. Our results are based on a new formula for f-invariant entropy.

On characters of Chevalley groups vanishing at the non-semisimple elements

Let [Formula: see text] be a finite simple group of Lie type. In this paper, we study characters of [Formula: see text] that vanish at the non-semisimple elements and whose degree is equal to the order of a maximal unipotent subgroup of [Formula: see text]. Such characters can be viewed as a natural generalization of the Steinberg character. For groups [Formula: see text] of small rank we also determine the characters of this degree vanishing only at the non-identity unipotent elements.

Arithmeticity and topology of smooth actions of higher rank abelian groups

We prove that any smooth action of $\mathbb{Z}^{m-1}$, $m\ge 3$, on an $m$-dimensional manifold that preserves a measure such that all non-identity elements of the suspension have positive entropy is essentially algebraic, i.e., isomorphic up to a finite permutation to an affine action on the torus or on its factor by $\pm\mathrm{Id}$. Furthermore this isomorphism has nice geometric properties; in particular, it is smooth in the sense of Whitney on a set whose complement has arbitrarily small measure. We further derive restrictions on topology of manifolds that may admit such actions, for example, excluding spheres and obtaining lower estimate on the first Betti number in the odd-dimensional case.

Generalized Torsion in Knot Groups

Abstract In a group, a nonidentity element is called a generalized torsion element if some product of its conjugates equals the identity. We show that for many classical knots one can ûnd generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the (hyperbolic) knot 52, and algebraic knots in the sense of Milnor.

All-involution table algebras and finite projective spaces

Table algebras all of whose nonidentity basis elements are involutions (in the sense of Zieschang), which serve as a counterpoint to the generic Hecke algebras parametrized by Coxeter groups, are classified. If two-generated, they are the family Hn (for all n ≥ 3), which for suitable n arise from schemes defined by affine planes of order n − 1. Otherwise, the basis involutions correspond to the points of a finite projective space whose incidence geometry determines the algebra multiplication. This generalizes to table algebras a previous result of van Dam for association schemes. An algebraic characterization is also given.

Permutation invariant lattices

We say that a Euclidean lattice in Rn is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group Sn, i.e., if the lattice is closed under the action of some non-identity elements of Sn. Given a fixed element τ∈Sn, we study properties of the set of all lattices closed under the action of τ: we call such lattices τ-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio (2002, 2007), which we previously studied in Fukshansky and Sun (2014). Continuing our investigation, we discuss some basic properties of permutation invariant lattices, in particular proving that the subset of well-rounded lattices in the set of all τ-invariant lattices in Rn has positive co-dimension (and hence comprises zero proportion) for all τ different from an n-cycle.

CAMINA p-groups that are generalized Frobenius complements

Let $P$ be a Camina $p$-group that acts on a group $Q$ in such a way that $C_P (x) \subseteq P'$ for all nonidentity elements $x \in Q$. We show that $P$ must be isomorphic to the quaternion group $Q_8$. If $P$ has class $2$, this is a known result, and this paper corrects a previously published erroneous proof of the general case.

Extension of Hölder’s theorem in

We prove that if ${\rm\Gamma}$ is a subgroup of $\text{Diff}_{+}^{\,1+{\it\epsilon}}(I)$ and $N$ is a natural number such that every non-identity element of ${\rm\Gamma}$ has at most $N$ fixed points, then ${\rm\Gamma}$ is solvable. If in addition ${\rm\Gamma}$ is a subgroup of $\text{Diff}_{+}^{\,2}(I)$, then we can claim that ${\rm\Gamma}$ is meta-abelian.

Difference systems of sets and a collection of 3-subsets in a finite field of order p

A difference system of set (DSS) is a collection of t disjoint τi-subsets Qi, 0≤i≤t−1, of Zn such that every non-identity element of Zn appears at least ρ times in the multiset {a−b|a∈Qi,b∈Qj,0≤i,j≤t−1,i≠j}. A DSS is regular if τi is constant for 0≤i≤t−1, and a DSS is perfect if every element of Zn is contained exactly ρ times in the above multiset. In this paper, we consider a collection of 3-subsets of a finite field of a prime order p=ef+1 to be a DSS. We present a condition for which the collection forms a regular DSS and give a lower bound on the parameter ρ using cyclotomic numbers for e=3,4 and 6. For the same values of e, we also show a condition for which a collection of 3-subsets is a perfect DSS.

Residual Properties of Nilpotent Groups

Let π be a set of primes. Recall that a group G is said to be a residually finite π-group if for every nonidentity element a of G there exists a homomorphism of the group G onto some finite π-group such that the image of the element a differs from 1. A group G will be said to be a virtually residually finite π-group if it contains a finite index subgroup which is a residually finite π-group. Recall that an element g in G is said to be π-radicable if g is an m-th power of an element of G for every positive π-number m. Let N be a nilpotent group and let all power subgroups in N are finitely separable. It is proved that N is a residually finite π-group if and only if N has no nonidentity π-radicable elements. Suppose now that π does not coincide with the set Π of all primes. Let π 0 be the complement of π in the set Π. And let T be a π 0 component of N i.e. T be a set of all elements of N whose orders are finite π 0 -numbers. We prove that the following three statements are equivalent: (1) the group N is a virtually residually finite π-group; (2) the subgroup T is finite and quotient group N/T is a residually finite π-group; (3) the subgroup T is finite and T coincides with the set of all π-radicable elements of N.

On fixed points of elements in primitive permutation groups

The fixity of a finite permutation group is the maximal number of fixed points of a non-identity element. We study the fixity of primitive groups of degree n, showing that apart from a short list of exceptions, the fixity of such groups is at least n1/6. We also prove that there is usually an involution fixing at least n1/6 points.