Groups Of Odd Order Research Articles

The Yamabe Invariant for Non-Simply Connected Manifolds

The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is nonnegative for all closed simply connected manifolds of dimension ≥ 5. We extend this to show that Yamabe invariant is nonnegative for all closed manifolds of dimension ≥ 5 with fundamental group of odd order having all Sylow subgroups abelian. The main new geometric input is a way of studying the Yamabe invariant on Toda brackets. A similar method of proof shows that all closed oriented manifolds of dimension ≥ 5 with non-spin universal cover, with finite fundamental group having all Sylow subgroups elementary abelian, admit metrics of positive scalar curvature, once one restricts to the complement of manifolds whose homology classes are toral. The exceptional toral homology classes only exist in dimensions not exceeding the rank of the fundamental group, so this proves important cases of the Gromov-Lawson-Rosenberg Conjecture once the dimension is sufficiently large.

Regular representations of finite groups as automorphism groups of k‐tournaments

AbstractIt was shown by Babai and Imrich [2] that every finite group of odd order except $Z^2_3$ and $Z^3_3$ admits a regular representation as the automorphism group of a tournament. Here, we show that for k ≥ 3, every finite group whose order is relatively prime to and strictly larger than k admits a regular representation as the automorphism group of a k‐tournament. Our constructions are elementary, suggesting that the problem is significantly simpler for k‐tournaments than for binary tournaments. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 238–248, 2002

Odd order groups with the rewriting property Q 3

Let n be an integer greater than 1, and let G be a group. A subset {x 1, x 2, ..., x n } of n elements of G is said to be rewritable if there are distinct permutations \( \pi \) and \( \sigma \) of {1, 2, ..., n} such that¶¶\( x_{\pi(1)}x_{\pi(2)} \ldots x_{\pi(n)} = x_{\sigma(1)}x_{\sigma(2)} \ldots x_{\sigma(n)}. \)¶¶A group is said to have the rewriting property Q n if every subset of n elements of the group is rewritable. In this paper we prove that a finite group of odd order has the property Q 3 if and only if its derived subgroup has order not exceeding 5.

Induction and Character Correspondences in Groups of Odd Order

Let P be a Sylow p-subgroup of G. By Irrp′(G), we denote the set of irreducible characters of G which have degree not divisible by p. When G is a solvable group of odd order, M. Isaacs constructed a natural one-to-one correspondence *:Irrp′(G)→Irrp′(NG(P)) which depends only on G and P. In this paper, we show that if ξG=χ∈Irrp′(G), then (ξ*)NG(P)=χ*.

Classifying 2-arc-transitive graphs of order a product of two primes

A classification of all 2-arc-transitive graphs of order a product of two primes is given. Furthermore, it is shown that cycles and complete graphs are the only 2-arc-transitive Cayley graph of Abelian group of odd order.

The dimension subgroup conjecture holds for odd order groups

The dimension subgroup conjecture holds for odd order groups

Surgery and the spectrum of the Dirac operator

We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension at least 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for the Dirac spectrum which says that if one performs surgery of codimension at least 3 on a closed Riemannian spin manifold, then the Dirac spectrum changes arbitrarily little provided the metric on the manifold after surgery is chosen properly.

Transversals of additive Latin squares

LetA={a 1, …,a k} andB={b 1, …,b k} be two subsets of an Abelian groupG, k≤|G|. Snevily conjectured that, whenG is of odd order, there is a permutationπ ∈S ksuch that the sums α i +b i , 1≤i≤k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even whenA is a sequence ofk<|G| elements, i.e., by allowing repeated elements inA. In this last sense the result does not hold for other Abelian groups. With a new kind of application of the polynomial method in various finite and infinite fields we extend Alon’s result to the groups (ℤ p ) a and\(\mathbb{Z}_{p^a } \) in the casek<p, and verify Snevily’s conjecture for every cyclic group of odd order.

Actions of finite p-groups on homology manifolds

Let a cyclic group of odd prime order p act on a ℤ(p)-Poincaré duality space X. We prove a relation between the Witt classes associated to the [ ]p-cohomology rings of the fixed point set of this action and of X. This is applied to show a similar result for actions of finite p-groups on ℤ(p)-homology manifolds.

A Note on Finite Metabelian Groups of Wielandt Length Two

The class of groups of Wielandt length one has been extensively studied and recently the class of groups of Wielandt length two has been studied, in particular nilpotent groups of odd order by E.A. Ormerod and supersoluble groups of order prime to six by A. Ali. In this paper we consider metabelian groups of odd order and provide characterizations for those groups with nilpotent residual either a Hall subgroup or of prime power order.

Equivariant periodicity for abelian group actions

For a manifold M, the structure set SOM; rel qU is the collection of manifolds homo- topy equivalent to M relative to the boundary. Siebenmann (11) showed that in the topological category, the structure set is 4-periodic in the sense that SOM; rel qUG SOM D 4 ; rel qU up to a copy of Z. The periodicity has been extended in (27) to topological manifolds with homotopi- cally stratified group actions by odd order groups, with D 4 replaced by the unit ball of any 4- fold permutation representation. In this paper, we extend such equivariant periodicity to the case that the group is compact abelian, and D 4 is replaced by the unit ball of twice of any

On (k, l)-sets in cyclic groups of odd prime order

Let A be a finite Abelian group written additively. For two positive integers k, l with k ≠ l, we say that a subset S ⊂ A is of type (k, l) or is a (k, l) -set if the equation x1 + x2 + … + xk − xk+1−… − xk+1 = 0 has no solution in the set S. In this paper we determine the largest possible cardinality of a (k, l)-set of the cyclic group ℤP where p is an odd prime. We also determine the number of (k, l)-sets of ℤp which are in arithmetic progression and have maximum cardinality.

Cocyclic Orthogonal Designs and the Asymptotic Existence of Cocyclic Hadamard Matrices and Maximal Size Relative Difference Sets with Forbidden Subgroup of Size 2

This paper contains a discussion of cocyclic Hadamard matrices, their associated relative difference sets, and regular group actions. Nearly all central extensions of the elementary abelian 2-groups by Z2 are shown to act regularly on the associated group divisible design of the Sylvester Hadamard matrices. Cocyclic orthogonal designs are then introduced, and the construction and classification questions for cocyclic M-concordant systems of orthogonal designs are addressed. (M-concordance generalises the concepts of amicability and anti-amicability.) We give an algebraic procedure for constructing and classifying these designs when each indeterminate is constrained to appear just once in each row and column of the orthogonal designs. This paper also gives a general (but apparently not comprehensive) method for constructing systems with no zero entries. In particular, we obtain a cocyclic pair of amicable OD(16; 8, 8). Using this pair of designs, we prove there is a cocyclic Hadamard matrix of order 2ts for any odd integer s>1 and any t⩾⌊8log2s⌋. A consequence of our argument is the theorem, “Let S be any group of odd order s containing k prime order subgroups Si of S such that (1) for i=0, …, k−1, the sets Ui=Si+1Si+2…Sk are subgroups of S, (2) for i=0, …, k−1, Si∩Ui=〈1〉, and (3) U0=S. Let T be any group of order 2t+1 containing a central involution z such that (1) T/〈z〉 is elementary abelian, and (2) the largest elementary abelian direct factor of T has rank between 4log2s and t−2−4log2s. Then there is a normal relative difference set of size 2ts with forbidden subgroup 〈z〉 in the group T×S.” The condition on S holds for any group of odd square-free order or any direct product of odd order elementary abelian groups.

Structure Theorems for Group Ring Codes with an Application to Self-Dual Codes

Using ideas from the cohomology of finite groups, an isomorphism is established between a group ring and the direct sum of twisted group rings. This gives a decomposition of a group ring code into twisted group ring codes. In the abelian case the twisted group ring codes are (multi-dimensional) constacyclic codes. We use the decomposition to prove that, with respect to the Euclidean inner product, there are no self-dual group ring codes when the group is the direct product of a 2-group and a group of odd order, and the ring is a field of odd characteristic or a certain modular ring. In particular, there are no self-dual abelian codes over the rings indicated. Extensions of these results to non-Euclidean inner products are briefly discussed.

Remarks on normal bases

We prove that any Galois extension of a commutative ring with a normal basis and abelian Galois group of odd order has a self-dual normal basis. We apply this result to get a very simple proof of nonexistence of normal bases for certain extensions which a

Equivariant deformations of manifolds and real representations

In the paper we give a partial answer to the following question: let G be a finite group acting smoothly on a compact (smooth) manifold M , such that for each isotropy subgroup H of G the submanifold M fixed by H can be deformed without fixed points; is it true that then M can be deformed without fixed points G-equivariantly? The answer is no, in general. It is yes, for any G-manifold, if and only if G is the direct product of a 2-group and an odd-order group.

On generic cyclic polynomials of odd prime degree

Using Cohen's construction of defining polynomials for a cyclic group of odd prime order, we define a polynomial with some parameters which generates cyclic extensions of a given odd prime degree, and prove it to be generic in the sense as defined below.

On Restricted Sums

Let G be an abelian group. For a subset A ⊂ G, denote by 2 ∧ A the set of sums of two different elements of A. A conjecture by Erdős and Heilbronn, first proved by Dias da Silva and Hamidoune, states that, when G has prime order, [mid ]2 ∧ A[mid ] [ges ] min([mid ]G[mid ], 2[mid ]A[mid ] − 3).We prove that, for abelian groups of odd order (respectively, cyclic groups), the inequality [mid ]2 ∧ A[mid ] [ges ] min([mid ]G[mid ], 3[mid ]A[mid ]/2) holds when A is a generating set of G, 0 ∈ A and [mid ]A[mid ] [ges ] 21 (respectively, [mid ]A[mid ] [ges ] 33). The structure of the sets for which equality holds is also determined.

The Borsuk-Ulam property for cyclic groups

An orthogonal representation $V$ of a group $G$ is said to have the Borsuk-Ulam property if the existence of an equivariant map $f:S(W) \rightarrow S(V)$ from a sphere of representation $W$ into a sphere of representation $V$ implies that $\dim W \leq \dim V$. It is known that a sufficient condition for $V$ to have the Borsuk-Ulam property is the nontriviality of its Euler class ${\text {\bf e}}(V)\in H^{*} (BG;\mathcal R)$. Our purpose is to show that ${\text {\bf e}}(V) \neq 0 $ is also necessary if $G$ is a cyclic group of odd and double odd order. For a finite group $G$ with periodic cohomology an estimate for $G$-category of a $G$-space $X$ is also derived.

On the Normality of Cayley Digraphs of Valency 2 on Non-Abelian Group of Odd Order

On the Normality of Cayley Digraphs of Valency 2 on Non-Abelian Group of Odd Order