Group Of Odd Prime Order Research Articles

The Spectrum of Group-Based Complete Latin Squares

We construct sequencings for many groups that are a semi-direct product of an odd-order abelian group and a cyclic group of odd prime order. It follows from these constructions that there is a group-based complete Latin square of order $n$ if and only if $n \in \{ 1,2,4\}$ or there is a non-abelian group of order $n$.

The η invariant of the Atiyah–Patodi–Singer operator on compact flat manifolds

Let \({\mathcal{D}}\) be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold M. In continuation of our previous study, we deal with the problem of computing explicitly the η invariant η = η(M) for any orientable compact flat manifold M. After giving an explicit expression for η(s) in the case of cyclic holonomy group, we obtain a combinatorial formula that reduces the computation to the cyclic case. We illustrate the method by determining η(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ≥ 7, η(s) can be expressed as a multiple of Lχ(s), an L-function associated to a quadratic character mod p, while η(0) is a (non-zero) integral multiple of the class number h−p of the number field \({\mathbb Q(\sqrt {-p})}\) . In the case of metacyclic groups of odd order pq, with p, q primes, we show that η(0) is a rational multiple of h−p.

The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group \({\mathbb{Z}_p}\), where p is an odd prime. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p = n = 3. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.

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.

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.

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.

Noether's bound for polynomial invariants of finite groups

Let G be a finite group acting linearly on the polynomial algebra \(\Bbb C [V]\). We prove that if G is the semi-direct product of cyclic groups of odd prime order, then the algebra of polynomial invariants is generated by its elements whose degree is bounded by \({5 \over 8}|G|\). As a consequence we derive that \(\Bbb C [V]^G\) is generated by elements of degree \(\leqq {3 \over 4}|G|\) for any non-cyclic group G. This sharpens the improved bound for Noether's Theorem due to Schmid.

Finite groups with large automizers for their nonabelian subgroups

If a finite nonabelian group G has the property that $N_{G}(H)/C_{G}(H)\cong \hbox {Aut}\,(H)$ for every nonabelian subgroup H of G, then G is isomorphic to the holomorph of a group of odd prime order.

The cohomology representation of an action of 𝐶_{𝑝} on a surface

When a finite group G G acts on a surface S S , then H 1 ( S ; Z ) {H^1}(S;\,{\mathbf {Z}}) posseses naturally the structure of a Z G {\mathbf {Z}}G -module with invariant symplectic inner product. In the case of a cyclic group of odd prime order we describe explicitly this symplectic inner product space in terms of the fixed-point data of the action.

Construction of group actions on four-manifolds

It is shown that any cyclic group of odd prime order acts on any closed, simply connected topological 4 4 -manifold, inducing the identity on integral homology. The action is locally linear except perhaps at one isolated fixed point. In the case of primes greater than three a more careful argument is used to show that the action can be constructed to be locally linear.

The topological spherical space form problem—II existence of free actions

RECENT advances in calculation of projective class groups and of surgery obstruction groups lead us to hope that it will shortly be possible to give a fairly complete account of the classification of free actions of finite groups on spheres. In the present paper, we determine which groups can so act, thus solving a problem of several years’ standing. Further, we show that these actions can be taken to be smooth actions on smooth homotopy spheres. Previously known results can be summarised as follows, where we say the finite group 7~ satisfies the “pq-condition” (p, q primes not necessarily distinct) if all subgroups if v of order pq are cyclic. 0.1. (Cartan and Eilenberg[3]). If rr acts freely on S”-‘, it has periodic cohomology with minimum period dividing n. Moreover, P has periodic cohomology if and only if it satisfies all p2-conditions. And the p* condition is equivalent to the Sylow p-subgroup zrr, of r being cyclic or perhaps (if p = 2) generalised quaternionic. 0.2. (Wolf [19]). If 7~ acts freely and orthogonally on a sphere, it satisfies all pq-conditions. Conversely, if r is soluble and satisfies all pq-conditions, free orthogonal actions exist. However, for rr non-soluble, the only non-cyclic composition factor allowed is the simple group of order 60. 0.3. (Milnor [9], see also Lee [8]). If 7~ acts freely on any sphere, it satisfies all 2p-conditions. 0.4. (Petrie [I 11). Any extension of a cyclic group of odd order m by a cyclic group of odd prime order q prime to m can act freely on S*“-‘. Petrie’s result shows that pq-conditions are not all necessary for free topological actions. it is therefore not so surprising that