Non-identity Element Research Articles

Lattices in one-dimensional quaternionic linear space that are invariant with respect to finite linear groups

i = j = k = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j. We identify C with the subfield R+ Ri ofH. Let V be a right n-dimensional linear space over H. A finite-order element of GL(V ) is said to be a reflection if the fixed-point space of the element is (n− 1)-dimensional. All finite subgroups of GL(V ) generated by reflections are classified in [1]. Aiming at the extension to the quaternion case of the classification of the discrete groups generated by affine Hermitian reflections, which was obtained in [2], we arrive at the problem of describing, up to a homothety (i.e., up to multiplication by a nonzero element of H), the lattices in V that are invariant with respect to the finite subgroups of GL(V ) generated by reflections (by a lattice we mean a free subgroup of rank 4n in the additive group of V ). In this note, such a description is given for n = 1. This description is heavily used in the investigation of the general case, which will be treated elsewhere. Thus, in what follows, we assume that V is H with right multiplication by constants End(V ) is H acting on V by left multiplication, and GL(V ) is the multiplicative group H∗ of the skew field H. All nonidentity elements of finite order inH∗ are reflections. The description of finite subgroups ofH∗ is well known (see, e.g., [3]); it is given by the following proposition (below by 〈M〉 we denote the subgroup ofG generated by some subsetM ⊆ G). Proposition 1. Up to conjugacy, the finite subgroups of H∗ are exhausted by the following pairwise nonconjugate subgroups:

An Application of Cyclotomic Polynomial to Factorization of Abelian Groups

If a finite abelian group G is a direct product of its subsets such that G = A1···Ai···An, G is said to have the Hajos-n-proprty if it follows that one of these subsets, say Ai is periodic, meaning that there exists a nonidentity element g in G such that gAi = Ai . Using some properties of cyclotomic polynomials, we will show that the cyclic groups of orders pα and groups of type (p2,q2) and (pα,pβ) where p and q are distinct primes and α, β integers ≥ 1 have this property.

A CONDITION IN FINITE SOLVABLE GROUPS RELATED TO CYCLIC SUBGROUPS

AbstractIn this paper, we classify the finite groups belonging to the class of cyclic-transitive groups. A group G is said to be cyclic-transitive if the following condition holds: if x, y, z are nonidentity elements of G such that 〈x,y〉 and 〈y,z〉 are both cyclic, then 〈x,z〉 is also cyclic.

Hamiltonian cycles in the generating graphs of finite groups

For a finite group G let Γ(G) denote the graph defined on the non-identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. In this paper it is shown that the graph Γ(G) contains a Hamiltonian cycle for many finite groups G.

On the minimal norm of a non-regular generalized character of an arbitrary finite group

We prove that, for any generalized character θ, other than a multiple of the regular character, of any finite group G, we have ∑ g ∊ G # | θ ( g ) | 2 ⩾ | G | χ ( 1 ) − 1 , where χ is an irreducible character of maximal degree of G and G# is the set of non-identity elements of G. We also determine exactly when equality is achieved.

Group Gradings in Incidence Algebras

If X is a bounded countable locally finite partially ordered set, R an integral domain, and G a group having the property that no non-identity element has order a unit of R. Then it is shown that any G-grading of the incidence algebra I(X, R) is equivalent to a good grading. Further, an example is given showing that not all group gradings of incidence algebras are equivalent to good gradings.

Connectivity of Random Cubic Sum Graphs

Consider the set $\mathcal{SG}(Q_k)$ of all graphs whose vertices are labeled with nonidentity elements of the group $Q_k=\mathbb{Z}_2^k$ so that there is an edge between vertices with labels a and b if and only if the vertex labeled $a+b$ is also in the graph. Note that edges always appear in triangles since $a+b=c$, $b+c=a$, and $a+c=b$ are equivalent statements for $Q_k$. We define the random cubic sum graph $\mathcal{SG}(Q_k,p)$ to be the probability space over $\mathcal{SG}(Q_k)$ whose vertex sets are determined by $\Pr[x\in V]=p$ with these events mutually independent. As p increases from 0 to 1, the expected structure of $\mathcal{SG}(Q_k,p)$ undergoes radical changes. We obtain thresholds for some graph properties of $\mathcal{SG}(Q_k,p)$ as $k\rightarrow\infty$. As with the classical random graph, the threshold for connectivity coincides with the disappearance of the last isolated vertex.

Simultaneous Ping-Pong Partners in PSL n (ℤ)

We show that for any finite set F of nonidentity elements in PSL n (ℤ) for n ≥ 3, consisting of hyperbolic, finite order, or unipotent elements, there exists an element g of infinite order in PSL n (ℤ) such that for any h ∈ F, the subgroup ⟨ g, h ⟩ generated by g and h is canonically isomorphic to the free product ⟨ g ⟩ * ⟨ h ⟩. We also show that the set of such elements in PSL n (ℤ) is Zariski dense in PSL n (ℝ).

Isomorphism Types of Maximal Cofinitary Groups

AbstractA cofinitary group is a subgroup of Sym(ℕ) where all nonidentity elements have finitely many fixed points. A maximal cofinitary group is a cofinitary group, maximal with respect to inclusion. We show that a maximal cofinitary group cannot have infinitely many orbits. We also show, using Martin's Axiom, that no further restrictions on the number of orbits can be obtained. We show that Martin's Axiom implies there exist locally finite maximal cofinitary groups. Finally we show that there exists a uniformly computable sequence of permutations generating a cofinitary group whose isomorphism type is not computable.

N-extended quasi-Baer rings

A ring with unit is said to be an n-extended right (principally) quasi-Baer ring if, for any proper (principal) right ideals I 1, …, I n , where n ≥ 2, the right annihilator of the product I 1 … I n is generated by an idempotent. A ring with unit is said to be an n-extended right (left) PP ring if the right (left, respectively) annihilator of the product x 1 … x n , where n ≥ 2, is generated by an idempotent for any nonidentity elements x 1, …, x n . The behavior of n-extended right (principally) quasi-Baer rings and right PP rings under various constructions and extensions is studied. These classes of rings are closed with respect to direct products and Morita equivalences. Examples illustrating the theory and outlining its frontiers are presented.

Distance-transitive graphs admit semiregular automorphisms

A distance-transitive graph is a graph in which for every two ordered pairs of vertices ( u , v ) and ( u ′ , v ′ ) such that the distance between u and v is equal to the distance between u ′ and v ′ there exists an automorphism of the graph mapping u to u ′ and v to v ′ . A semiregular element of a permutation group is a non-identity element having all cycles of equal length in its cycle decomposition. It is shown that every distance-transitive graph admits a semiregular automorphism.

Perfect difference systems of sets and Jacobi sums

A perfect ( v , { k i ∣ 1 ≤ i ≤ s } , ρ ) difference system of sets (DSS) is a collection of s disjoint k i -subsets D i , 1 ≤ i ≤ s , of any finite abelian group G of order v such that every non-identity element of G appears exactly ρ times in the multiset { a − b ∣ a ∈ D i , b ∈ D j , 1 ≤ i ≠ j ≤ s } . In this paper, we give a necessary and sufficient condition in terms of Jacobi sums for a collection { D i ∣ 1 ≤ i ≤ s } defined in a finite field F q of order q = e f + 1 to be a perfect ( q , { k i ∣ 1 ≤ i ≤ s } , ρ ) -DSS, where each D i is a union of cyclotomic cosets of index e (and the zero 0 ∈ F q ). Also, we give numerical results for the cases e = 2 , 3 , and 4.

Generalizations of Partial Difference Sets from Cyclotomy to Nonelementary Abelian $p$-Groups

A partial difference set having parameters $(n^2, r(n-1), n+r^2-3r,r^2-r)$ is called a Latin square type partial difference set, while a partial difference set having parameters $(n^2, r(n+1), -n+r^2+3r,r^2+r)$ is called a negative Latin square type partial difference set. In this paper, we generalize well-known negative Latin square type partial difference sets derived from the theory of cyclotomy. We use the partial difference sets in elementary abelian groups to generate analogous partial difference sets in nonelementary abelian groups of the form $(Z_p)^{4s} \times (Z_{p^s})^4$. It is believed that this is the first construction of negative Latin square type partial difference sets in nonelementary abelian $p$-groups where the $p$ can be any prime number. We also give a generalization of subsets of Type Q, partial difference sets consisting of one fourth of the nonidentity elements from the group, to nonelementary abelian groups. Finally, we give a similar product construction of negative Latin square type partial difference sets in the additive groups of $(F_q)^{4t+2}$ for an integer $t \geq 1$. This construction results in some new parameters of strongly regular graphs.

On the asymmetric complexity of the group-intersection problem

Motivated by the question of determining the difference in complexity between the graph-isomorphism and graph-automorphism problems, we study the relationship between the coset-intersection (C-INT) and group-intersection (G-INT) problems in permutation groups. We prove that G-INT is related to C-INT asymmetrically under Ogiwara and Watanabe's notions of left and right sets. This asymmetry suggests that, for given two permutation groups that intersect nontrivially, finding the least nonidentity element in the intersection is likely to be easier than finding the largest nonidentity element in the intersection under any lexicographic ordering. Also as a consequence, unless C-INT (resp., G-INT) ∈P, C-INT (resp., G-INT) is not many-one reducible in polynomial time to any sparse set.

Conjugacy in lattice-ordered groups and right ordered groups

We prove that if Gj are right orderable groups with non-trivial cyclic subgroups Cj (j = 1,2), then the (group) free product of G1 and G2 with C1 and C2 amalgamated is right orderable. We will deduce this from our main theorem that every lattice-ordered group can be embedded in one in which there are precisely four conjugacy classes. As a consequence of our method, we also show that every right ordered group can be embedded in a right ordered group in which any two non-identity elements are conjugate.

Distinguishability of Locally Finite Trees

The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$, $k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees without pendant vertices are shown to be 2-distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results are obtained for the distinguishing chromatic number, namely the least positive integer $n$ such that the function $f$ is also a proper vertex-coloring.

The exact spread of the Mathieu group M 11

Following [J. I. Brenner and J. Wiegold. Two-generator groups, I. Michigan Math. J. 22 (1975), 53–64.], we define the exact spread of a group G to be the greatest integer r for which, given any r non-identity elements of G, one can find y ∈ G such that 〈xi = y〉 = G for i = 1, 2, …, r. In this paper, we determine the exact spread of M11, the smallest sporadic simple group of Mathieu.

On strong reality of the unipotent Lie-type subgroups over a field of characteristic 2

A group G is called strongly real if its every nonidentity element is strongly real, i.e. conjugate with its inverse by an involution of G. We address the classical Lie-type groups of rank l, with l ≤ 4 and l ≤ 13, over an arbitrary field, and the exceptional Lie-type groups over a field K with an element η such that the polynomial X 2 + X + η is irreducible either in K[X] or K 0[X] (in particular, if K is a finite field). The following question is answered for the groups under study: What unipotent subgroups of the Lie-type groups over a field of characteristic 2 are strongly real?

All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism

The polycirculant conjecture states that every transitive 2-closed permutation group of degree at least two contains a nonidentity semiregular element, that is, a nontrivial permutation whose cycles all have the same length. This would imply that every vertex-transitive digraph with at least two vertices has a nonidentity semiregular automorphism. In this paper we make substantial progress on the polycirculant conjecture by proving that every vertex-transitive, locally-quasiprimitive graph has a nonidentity semiregular automorphism. The main ingredient of the proof is the determination of all biquasiprimitive permutation groups with no nontrivial semiregular elements.

On generalized characters of nilpotent groups

We extend the results of [G.R. Robinson, More on bounds on norms of generalized characters with applications to p-local bounds and blocks, Bull. London Math. Soc. 37 (4) (2005) 555–565] to a general finite nilpotent group, rather than a p-group. This enables us to obtain bounds on the number of complex irreducible characters of a finite group G which do not vanish identically on all non-identity elements of a nilpotent subgroup N.