Cyclic Group Of Prime Order Research Articles

On the group 𝑆𝑆𝐹(𝐺),𝐺 a cyclic group of prime order

We extend the definition of the obstruction group SSF ( G ) {\text {SSF}}(G) in the case where G G is a cyclic group of prime order. We show that an endomorphism of a free Z G ZG -module is a direct summand of a virtual permutation if its characteristic polynomial has the appropriate form. Among these endomorphisms the virtual permutations are detected by K 0 {K_0} . The main application is in detecting Morse-Smale isotopy classes.

Connected Cayley graphs of semi-direct products of cyclic groups of prime order by abelian groups are hamiltonian

In this paper it is shown that every connected Cayley graph of a semi-direct product of a cyclic group of prime order by an abelian group is hamiltonian. In particular, every connected Cayley graph of a group G is hamiltonian provided that G is of order greater than 2 and it contains a normal cyclic subgroup N of prime order such that the quotient group G/ N is abelian and its order is relatively prime to that of N.

Hamiltonian circuits in Cayley graphs

The following result is proved: If either G is a finite abelian group or a semidirect product of a cyclic group of prime order by a finite abelian group of odd order, then every connected Cayley graph of G is hamiltonian.

On Modular Representation Algebras and a Class of Matrix Algebras

AbstractLet G be a cyclic group of prime order p and K a field of characteristic p. The set of classes of isomorphic indecomposable (K, G)-modules forms a basis over the complex field for an algebra p (Green, 1962) with addition and multiplication being derived from direct sum and tensor product operations.Algebras n with similar properties can be defined for all n ≥ 2. Each such algebra is isomorphic to a matrix algebra Mn of n × n matrices with complex entries and standard operations. The characters of elements of n are the eigenvalues of the corresponding matrices in Mn.

Recurrence relations in a modular representation algebra

In 1978 Almkvist and Fossum examined the decomposition of the exterior powers of basis modules in the modular representation algebra of a cyclic group of prime order. In particular they developed an isomorphism between these exterior powers and terms of binomial coefficient type in the algebra.We derive several recurrence relations for these terms.

TWO THEOREMS FROM THE THEORY OF PERIODIC TRANSFORMATIONS

Let be a connected paracompact Hausdorff space, freely acted on by a cyclic group of prime order with generator . Let be a continuous mapping of into a topological manifold of dimension . Put . If is orientable over , for , and has zero image, then, for weakly locally contractible, . If, in addition, is an -dimensional topological manifold, then . For , suppose and , while is a connected compact closed manifold of dimension with a free involution . Let , and suppose is a monomorphism. Then .Bibliography: 5 titles.

A GENERALIZATION OF THE BORSUK-ULAM THEOREM

Let be a connected paracompact Hausdorff space, acted on without fixed points by a cyclic group of prime order . For any continuous mapping let where is a generator of . Suppose for , and is a compact -orientable topological manifold of dimension . If the mapping has zero image, then the cohomological dimension over of the set is at least . Furthermore, if is a generalized manifold of dimension , and , then .Bibliography: 8 titles.

A Solution of a Problem of Steenrod for Cyclic Groups of Prime Order

Given a $Z[G]$ module A, we will say a simply connected CW complex X is of type (A, n) if X admits a cellular G action, and ${\tilde H_i}(X) = 0,i \ne n,{H_n}(X) \simeq A$ as $Z[G]$ modules. In [5], R. Swan considers the problem posed by Steenrod of whether or not there are finite complexes of type (A, n) for all finitely generated A and finite G. Using an invariant defined in terms of ${G_0}(Z[G])$, solutions were obtained for $A = {Z_p}$ (p-prime) and $G \subseteq \operatorname {Aut}\;({Z_p})$. The question of infinite complexes of type (A, n) was left open. In this paper we obtain the following complete solution for $Z[{Z_p}]$ modules: There are complexes of type $(A,n)\;(n \geqslant 3)$, and there are finite complexes of type (A, n) if and only if the invariant which corresponds to Swan’s invariant for these modules vanishes.

A solution of a problem of Steenrod for cyclic groups of prime order

Given a Z [ G ] Z[G] module A, we will say a simply connected CW complex X is of type (A, n) if X admits a cellular G action, and H ~ i ( X ) = 0 , i ≠ n , H n ( X ) ≃ A {\tilde H_i}(X) = 0,i \ne n,{H_n}(X) \simeq A as Z [ G ] Z[G] modules. In [5], R. Swan considers the problem posed by Steenrod of whether or not there are finite complexes of type (A, n) for all finitely generated A and finite G. Using an invariant defined in terms of G 0 ( Z [ G ] ) {G_0}(Z[G]) , solutions were obtained for A = Z p A = {Z_p} (p-prime) and G ⊆ Aut ( Z p ) G \subseteq \operatorname {Aut}\;({Z_p}) . The question of infinite complexes of type (A, n) was left open. In this paper we obtain the following complete solution for Z [ Z p ] Z[{Z_p}] modules: There are complexes of type ( A , n ) ( n ⩾ 3 ) (A,n)\;(n \geqslant 3) , and there are finite complexes of type (A, n) if and only if the invariant which corresponds to Swan’s invariant for these modules vanishes.

Equivariant Bordism of Maps

This note computes the bordism classification of equivariant maps between closed manifolds with action of a cyclic group of prime order.

ACTIONS OF FINITE CYCLIC GROUPS ON QUASICOMPLEX MANIFOLDS

In this paper a classification is given of actions of finite cyclic groups on quasicomplex manifolds in terms of the invariants of cobordism theory. Moreover, the methods of the paper allow one to understand the geometric nature of known results of a series of authors on actions of cyclic groups of prime order. Bibliography: 11 items.

Equivariant bordism of maps

This note computes the bordism classification of equivariant maps between closed manifolds with action of a cyclic group of prime order.

The Augmentation Terminals of Certain Locally Finite Groups

Let G be a group and ZG be the integral group ring of G. We shall write 𝔤 for the augmentation ideal of G; that is to say, the kernel of the homomorphism of ZG onto Z which sends each group element to 1. The powers gλ of 𝔤 are defined inductively for ordinals λ by 𝔤λ = 𝔤μ𝔤, if λ = μ + 1, and otherwise. The first ordinal λ for which gλ = 𝔤λ+1 is called the augmentation terminal or simply the terminal of G. For example, if G is either a cyclic group of prime order or else isomorphic with the additive group of rational numbers then gn > 𝔤ω = 0 for all finite n, so that these groups have terminal ω.The groups with finite terminal are well-known and easily described. If G is one such, then every homomorphic image of G must also have finite terminal.

The effect of an involution in $$\widetilde{K}^0 $$ (ZG)

The involution in the Grothendieck group of the group ring of a finite cyclic group of prime order p, induced by the transition to the contragredient module is identical to complex conjugation followed by the automorphism x → x−1 in the ideal class group of the cyclotomic field of order p.

Steenrod operations and transfer

In this note I show that for the cohomology of a group with coefficients in Z/pZ, the transfer homomorphism commutes with the Steenrod reduced power operations. Suppose that G is a group and H a subgroup of finite index. Let P denote the cyclic group of prime order p considered as a subgroup of Sp, the symmetric group on p symbols. As in [2 ], we denote by S 5 G, the semi direct product of the permutation group S with the group GP where the former acts on the latter by permuting the factors. Then P X G, for example, may be considered a subgroup of P S G by imbedding G in GP via the p-fold diagonal map. If this is done, the basic Steenrod construction may be described simply: Given a E Hq(G, Z/pZ), form the element 1 5 a E Hvq(P S G, Z/pZ) as follows. Let X be a G-projective resolution of Z and suppose f cHomG(Xq, Z/pZ) represents a. Let W be a P-projective resolution of Z with augmentation e. Then W0XP becomes a P 5 G projective resolution of Z and e0fP is a PS G homomorphism of W0XP into Z 0 (Z/p)P zZ/pZ which is in fact a cocycle whose cohomology classdenoted by 1 5 a-depends only on a. (More generally, we may replace P by any subgroup S of Sp, but then it is necessary to include a sign in the action of S on ZO (Z/pZ)P.) Denote by P(a) =PG(a) CHPq(P X G, Z/pZ) the restriction of 1 5 a to the subgroup P X G. P(a) is the basic object from which the reduced powers are constructed. Suppose next that j-Hq (H, Z/pZ). Let T= fr} be a left transversal of H in G. Since H is a subgroup of G, X is also an H-projective resolution of Z and if gCHomH(Xq, Z/pZ) represents 3, then 17ET 'rg represents trH G(O). (See [1, Chapter XII, ?8].) We wish to study P(tr /) =res(1 5 (tr /)). As above, 1 5 tr / is represented by

Integral representations of dihedral groups of order 2𝑝

Introduction. Information about the integral representations of finite groups has been obtained to varying extents. For Z the ring of rational integers and G the cyclic group of prime order, the ZG-modules were studied by Diederichsen [3] and Reiner [11], who showed that there were finitely many indecomposable ZG-modules and determined them completely. The finiteness of the number of indecomposables in the case where G is cyclic of order p2 was shown, for p = 2, by Troy [16] and for any p by Heller and Reiner [5] and by Knee [8], while Oppenheim [10] and Knee [8] established the finiteness of the number of indecomposables for G cyclic, of square free order. Heller and Reiner [5; 6] and Jones [7] established that the number of indecomposable ZG-modules is finite if and only if all p-Sylow subgroups of G are cyclic of order at most p2. Here, as well as throughout this paper, we shall mean by a ZG-module one which is finitely generated and Z-free. In this paper we shall classify all finitely-generated S-free SG-modules where G is the dihedral group of order 2p, p an odd prime, and S is Z or Z2p the semilocal ring formed by the intersection of Zp and Z2, respectively the rings of p-integral and 2-integral elements in s the rational field. Z2p = {r/seQ: (s,2p) = 1}. Taking 0 to be a primitive pth root of unity, we shall denote by K = Q(9) the cyclotomic field of degree p — 1 over s and by K0 = Q(9 + 9~l) the real subfield of K. R0 and R shall be the integral closures of S in K0 and K, respectively. Letting I) denote the group of automorphisms of K with fixed field K0, we may form A the twisted group ring of h with coefficients in R. §1 of this paper is devoted to a characterization of R-projective A-modules of finite R-rank. The results of this section are then applied in the second section to show that there are precisely 7/t + 3 nonisomorphic, indecomposable SGmodules where h is the ideal class number of R0. In §3 it is shown that although a Krull-Schmidt theorem is not obtainable for SG-modules, invariants may be obtained which determine an SG-module up to Z2pG-isomorphism. The final section deals with projective SG-modules. Here an isomorphism is established

The cohomology of certain orbit spaces

Let (G, X) be a topological transformation group—or action—in which G is finite and X is locally compact. An important part of the cohomology of the orbit space X/G lies, so to speak, in the free part f of the action (i.e. the union of orbits of cardinality [G: l ] ) . The cohomology of i/G can be regarded as an H(G) -module. We shall exhibit a complete set of generators and relations for this module assuming G to be the direct product of cyclic groups of prime order p and X to be a generalized sphere over Zp (see [4, p. 404]). H will always denote cohomology with values in Zv. A useful device consists in relating the generators of H(G) to those of G.

On the Ring of Integers in an Algebraic Number Field as a representation Module of Galois Group

1. Introduction. It is known that there are only three rationally inequivalent classes of indecomposable integral representations of a cyclic group of prime order l. The representations of these classes are: (I) identical representation,(II) rationally irreducible representation of degree l – 1,(III) indecomposable representation consisting of one identical representation and one rationally irreducible representation of degree l-1 (F. E. Diederichsen [1], I. Reiner [2]).

Integral Representations of Cyclic Groups of Prime Order

Integral Representations of Cyclic Groups of Prime Order

Integral representations of cyclic groups of prime order

Integral representations of cyclic groups of prime order