Cyclic Group Of Prime Order Research Articles

On fixed-point-free automorphisms

Let R be a cyclic group of prime order which acts on the extraspecial group F in such a way that F=[F,R]. Suppose RF acts on a group G so that CG(F)=1 and (|R|,|G|)=1. It is proved that F(CG(R))⊆F(G). As corollaries to this, it is shown that the Fitting series of CG(R) coincides with the intersections of CG(R) with the Fitting series of G, and that when |R| is not a Fermat prime, the Fitting heights of CG(R) and G are equal.

Decomposing modular coinvariants

We consider the ring of coinvariants for a modular representation of a cyclic group of prime order p. We show that the classes of the terminal variables in the coinvariants have nilpotency degree p and that the coinvariants are a free module over the subalgebra generated by these classes. An incidental result we have is a description of a Gröbner basis for the Hilbert ideal and a decomposition of the corresponding monomial basis for the coinvariants with respect to the monomials in the terminal variables.

On *-clean group rings

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.

Exterior and symmetric powers of modules for cyclic 2-groups

We prove a recursive formula for the exterior and symmetric powers of modules for a cyclic 2-group. This makes computation straightforward. Previously, a complete description was only known for cyclic groups of prime order.

A remark on a conjecture of Derksen

Let V be a complex representation of a finite group G of order g. Derksen conjectured that the pth syzygies of the invariant ring Sym(V ) are generated in degrees ≤ (p+ 1)g. We point out that a simple application of the theory of twisted commutative algebras — using an idea due to Weyl — gives the weaker bound pg, almost for free. Fix a finite group G of order g. Let V be a finite dimensional complex representation of G and put R = Sym(V )G, which we regard as a graded ring. Let E ⊂ R be a homogeneous vector subspace generating R as an algebra, and put S = Sym(E), so that S → R is a surjection of graded algebras. The space Torp (R,C) is then a graded vector space, called the space of p-syzygies of R. We let sp(V ;E) be the maximal degree occurring in it. One can show that if E ⊂ E′ then sp(V ;E) ≤ sp(V ;E′). Furthermore, sp(V ;E) is independent of E if E is chosen to be minimal; we denote this common value by sp(V ). We then have the following conjecture of Derksen (see [D, Conj. 3] for a more precise version): Conjecture 1. We have sp(V ) ≤ (p+ 1)g for any V . Derksen [D, Thm. 2] proved the conjecture for p = 1, but the general case is open. To state our main result, we first introduce some notation. Let β(V ) be the minimal integer such that R is generated in degrees ≤ β(V ). Let β be the maximum value of β(V ) over all V , the so-called Noether number of G. Noether’s theorem [W, §3] states that β ≤ g. Let d1, . . . , dn be the degrees of the irreducible representations of G and let m be the sum of the di. Finally, put δp = (β − 1)g − (m− 1)βp; note that this is negative for p 0. We then have: Theorem 2. We have sp(V ) ≤ β2mp+ δp for any V . Although this is weaker than the conjecture, the bound is significant since it is independent of V . Using the fact that β and m are both bounded by g, we deduce the following corollary: Corollary 3. We have sp(V ) ≤ pg3 for any V . Remark 4. In fact, m and β are often strictly smaller than g, sometimes significantly so. We have m ≤ √ng, where n is the number of conjugacy classes in G. Typically, n is much smaller than g; for example, if G is a symmetric group then n = O(g ), for any > 0. The currently known bounds on β are not very sharp. A result of Cziszter–Domokos [CD] states that β < 1 2g unless G has a cyclic subgroup of index two, or is one of four exceptions. A conjecture of Pawale [W, Conj. 3.9] asserts that if G is the semi-direct product of cyclic groups of prime orders p and q, with q | p− 1, then β = p+ q − 1; note that m = p+ q − 1 in this case as well. When p and q are approximately equal, this gives m = β = O( √ g). Thus, in many case, our theorem gives a bound of the form sp(V ) = O(pg θ) with θ < 3. Remark 5. Theorem 2 (and our proof of it) is valid over any field of characteristic zero. Derksen also proposed his conjecture in positive characteristic not dividing g. It would be interesting if our proof could be adapted to work in this setting. We now prove the theorem. Put sp(V ) = sp(V ;E) where E = ⊕β i=1Ri. As discussed, we have sp(V ) ≤ sp(V ), so it suffices to bound the latter. The key result is the following lemma. Date: November 2, 2012. The author was supported by NSF fellowship DMS-0902661. 1

The Noether number of the non-abelian group of order $$3p$$ 3 p

It is proven that for any representation over a field of characteristic $$0$$ of the non-abelian semidirect product of a cyclic group of prime order $$p$$ and the group of order $$3$$ the corresponding algebra of polynomial invariants is generated by elements of degree at most $$p+2$$ . We also determine the exact universal degree bound for separating systems of polynomial invariants of this group in characteristic not dividing $$3p$$ .

Finitely Generated Groups G such that G/Z(G) ≈ C p × C p

Finite groups G such that G/Z(G) ≈ C 2 × C 2 where C 2 denotes a cyclic group of order 2 and Z(G) is the center of G were studied in [5] and were used to classify finite loops with alternative loop algebras. In this paper we extend this result to finitely generated groups such that G/Z(G) ≈ C p × C p where C p denotes a cyclic group of prime order p and provide an explicit description of all such groups.

Unary polynomials on a class of semidirect products of finite groups

We describe unary polynomial functions on nite groups G that are semidirect products of an elementary abelian group of exponent p and a cyclic group of prime order q, p ≠ q.

Invariants for the modular cyclic group of prime order via classical invariant theory

Let \mathbb F be any field of characteristic p . It is well-known that there are exactly p inequivalent indecomposable representations V_1,V_2,\dots,V_p of C_p defined over \mathbb F . Thus if V is any finite dimensional C_p -representation there are non-negative integers 0\leq n_1,n_2,\dots, n_k \leq p-1 such that V \cong \oplus_{i=1}^k V_{n_i+1} . It is also well-known there is a unique (up to equivalence) d+1 dimensional irreducible complex representation of SL _2(\mathbb C) given by its action on the space R_d of d forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring of C_p -invariants \mathbb F[ \oplus_{i=1}^k V_{n_i+1}]^{C_p} to the computation of the classical ring of invariants (or covariants) \mathbb C[R_1 \oplus (\oplus_{i=1}^k R_{n_i})]^{\mathrm {SL}_2(\mathbb C)} . This shows that the problem of computing modular C_p invariants is equivalent to the problem of computing classical SL _2(\mathbb C) invariants. This allows us to compute for the first time the ring of invariants for many representations of C_p . In particular, we easily obtain from this generators for the rings of vector invariants \mathbb F[m\,V_2]^{C_p} , \mathbb F[m\,V_3]^{C_p} and \mathbb F[m\,V_4]^{C_p} for all m \in \mathbb N . This is the first computation of the latter two families of rings of invariants.

Characterization of p-valenced schemes of order [formula omitted

In [P.-H. Zieschang, Association schemes in which the thin residue is a finite cyclic group, J. Algebra 324 (2010) 3572–3578], it was shown that association schemes are schurian if their thin residues are finite groups and have a distributive normal subgroup lattice. However, in the case that their thin residues are elementary abelian groups, they do not have a distributive normal subgroup lattice. Actually, they are not necessarily schurian due to the non-schurian quasi-thin scheme of order 28. In this paper, we investigate association schemes such that their thin residues are elementary abelian groups of order p2. It is shown that these schemes are schurian if the number of elements with valency p is 2p and the factor scheme over the thin residue is a cyclic group of prime order. We also construct non-schurian p-schemes of order p3.

On stable conjugacy of finite subgroups of the plane Cremona group, I

AbstractWe discuss the problem of stable conjugacy of finite subgroups of Cremona groups. We compute the stable birational invariant H 1(G, Pic(X)) for cyclic groups of prime order.

Automorphisms of global semigroups of cyclic groups of prime order.

Automorphisms of global semigroups of cyclic groups of prime order.

On Representations of Cyclic Groups Over a Ring of Integers

The purpose of this article to determine and classify the indecomposable RG-lattices, where R is a certain ring of algebraic integers, and G is a cyclic group of prime order.

Cohomology of toroidal orbifold quotients

Let φ : Z / p → GL n ( Z ) denote an integral representation of the cyclic group of prime order p. This induces a Z / p -action on the torus X = R n / Z n . The goal of this paper is to explicitly compute the cohomology groups H ⁎ ( X / Z / p ; Z ) for any such representation. As a consequence we obtain an explicit calculation of the integral cohomology of the classifying space associated to the family of finite subgroups for any crystallographic group Γ = Z n ⋊ Z / p with prime holonomy.

ON THE DENSITY OF DISCRIMINANTS OF ABELIAN EXTENSIONS OF A NUMBER FIELD

Let G = Cℓ × Cℓ denote the product of two cyclic groups of prime order ℓ, and let k be an algebraic number field. Let N(k, G, m) denote the number of abelian extensions K of k with Galois group G(K/k) isomorphic to G, and the relative discriminant 𝒟(K/k) of norm equal to m. In this paper, we derive an asymptotic formula for ∑m≤XN(k, G; m). This extends the result previously obtained by Datskovsky and Mammo.

Basic quasi-Hopf algebras over cyclic groups

Let m be a positive integer, not divisible by 2, 3, 5, 7. We generalize the classification of basic quasi-Hopf algebras over cyclic groups of prime order given in Etingof and Gelaki (2006) [11] to the case of cyclic groups of order m. To this end, we introduce a family of non-semisimple radically graded quasi-Hopf algebras A ( H , s ) , constructed as subalgebras of Hopf algebras twisted by a quasi-Hopf twist, which are not twist equivalent to Hopf algebras. Any basic quasi-Hopf algebra over a cyclic group of order m is either semisimple, or is twist equivalent to a Hopf algebra or a quasi-Hopf algebra of type A ( H , s ) .

Vector invariants for the two-dimensional modular representation of a cyclic group of prime order

In this paper, we study the vector invariants of the 2-dimensional indecomposable representation V 2 of the cyclic group, C p , of order p over a field F of characteristic p, F [ m V 2 ] C p . This ring of invariants was first studied by David Richman (1990) [20] who showed that the ring required a generator of degree m ( p − 1 ) , thus demonstrating that the result of Noether in characteristic 0 (that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group) does not extend to the modular case. He also conjectured that a certain set of invariants was a generating set with a proof in the case p = 2 . This conjecture was proved by Campbell and Hughes (1997) in [3]. Later, Shank and Wehlau (2002) in [24] determined which elements in Richman's generating set were redundant thereby producing a minimal generating set. We give a new proof of the result of Campbell and Hughes, Shank and Wehlau giving a minimal algebra generating set for the ring of invariants F [ m V 2 ] C p . In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for F [ m V 2 ] C p . Further, our results provide a procedure for finding an explicit decomposition of F [ m V 2 ] into a direct sum of indecomposable C p -modules. Finally, noting that our representation of C p on V 2 is as the p-Sylow subgroup of SL 2 ( F p ) , we describe a generating set for the ring of invariants F [ m V 2 ] SL 2 ( F p ) and show that ( p + m − 2 ) ( p − 1 ) is an upper bound for the Noether number, for m > 2 .

On the Automorphic Chromatic Index of a Graph

In this paper we define the automorphic H-chromatic index of a graph Γ as the minimum integer m for which Γ has a proper edge-coloring with m colors which is preserved by a given automorphism group H of Γ. After the description of some properties, we determine upper bounds for this index when H is a cyclic group of prime order. We also show that these upper bounds are best possible in a number of instances.

The diameter of a random Cayley graph of ℤ q

Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k, we prove that the diameter of said graph is asymptotically (in q) of order k √ q. The same also holds when the generating set is taken to be a symmetric set of size 2k.

Lexsegment and Gotzmann ideals associated with the diagonal action of $${\mathbb Z/p}$$

We consider a diagonal action of a cyclic group of prime order on a polynomial ring F[x1, . . . , xn]. We give a description of the actions for which the corresponding Hilbert ideal is Gotzmann when n = 2. Nevertheless, we show that there is a separating set of invariant monomials that generates a proper lexsegment ideal in the polynomial ring for all n. As well, we provide an algorithm to compute this set.