Groups Of Odd Order Research Articles

Blocks of defect zero in finite groups with conjugate subgroups of a given prime order

Let [Formula: see text] be a finite group and [Formula: see text] be a prime. In this note, we show that if [Formula: see text] and all subgroups of [Formula: see text] of order [Formula: see text] are conjugate, then either [Formula: see text] has a [Formula: see text]-block of defect zero, or [Formula: see text] and [Formula: see text] is a direct product of a simple group [Formula: see text] and an odd order group. This improves one of our previous works.

On the realizable classes of the square root of the inverse different in the unitary class group

Let [Formula: see text] be a number field with ring of integers [Formula: see text] and let [Formula: see text] be a finite abelian group of odd order. Given a [Formula: see text]-Galois [Formula: see text]-algebra [Formula: see text], write [Formula: see text] for its trace map and [Formula: see text] for its square root of the inverse different, where [Formula: see text] exists by Hilbert’s formula since [Formula: see text] has odd order. The pair [Formula: see text] is locally [Formula: see text]-isometric to [Formula: see text] whenever [Formula: see text] is weakly ramified, in which case it defines a class in the unitary class group [Formula: see text] of [Formula: see text]. Here [Formula: see text] denotes the canonical symmetric bilinear form on [Formula: see text] defined by [Formula: see text] for all [Formula: see text]. We will study the set of all such classes and show that a subset of them forms a subgroup of [Formula: see text].

Formally dual subsets of cyclic groups of prime power order

We study the notion of formal-duality over finite cyclic groups of prime power order as introduced by Cohn, Kumar, Reiher and Sch\"urmann. We will prove that for any cyclic group of odd prime power order, as well as for any cyclic group of order $2^{2l+1}$, there is no primitive pair of formally-dual subsets. This partially proves a conjecture, made by the priorly mentioned authors, that the only cyclic groups with a pair of primitive formally-dual subsets are $\{0\}$ and $\mathbb{Z}/4\mathbb{Z}$.

On S-quasinormally embedded subgroups of finite groups

A subgroup H of a group G is said to be S-quasinormally embedded in G if for every Sylow subgroup P of H, there is an S-quasinormal subgroup K in G such that P is also a Sylow subgroup of K. Groups with certain S-quasinormally embedded subgroups of prime power order are studied. We prove Theorems 1.4, 1.5 and 1.6 of [10] remain valid if we omit the assumption that G is a group of odd order.

Augmentation quotients for burnside rings of generalized dihedral groups

Let H be a finite abelian group of odd order, D be its generalized dihedral group, i.e., the semidirect product of C2 acting on H by inverting elements, where C2 is the cyclic group of order two. Let Ω (D) be the Burnside ring of D, Δ(D) be the augmentation ideal of Ω (D). Denote by Δn(D) and Qn(D) the nth power of Δ(D) and the nth consecutive quotient group Δn(D)/Δn+1(D), respectively. This paper provides an explicit Z-basis for Δn(D) and determines the isomorphism class of Qn(D) for each positive integer n.

A quotient of the Artin braid groups related to crystallographic groups

Let n≥3. In this paper, we study the quotient group Bn/[Pn,Pn] of the Artin braid group Bn by the commutator subgroup of its pure Artin braid group Pn. We show that Bn/[Pn,Pn] is a crystallographic group, and in the case n=3, we analyse explicitly some of its subgroups. We also prove that Bn/[Pn,Pn] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of Bn/[Pn,Pn] with the conjugacy classes of the elements of odd order of the symmetric group Sn, and that the isomorphism class of any Abelian subgroup of odd order of Sn is realised by a subgroup of Bn/[Pn,Pn]. Finally, we discuss the realisation of non-Abelian subgroups of Sn of odd order as subgroups of Bn/[Pn,Pn], and we show that the Frobenius group of order 21, which is the smallest non-Abelian group of odd order, embeds in Bn/[Pn,Pn] for all n≥7.

Finite dimensional Nichols algebras over certain p-groups

ABSTRACTWe investigate pointed Hopf algebras over finite nilpotent groups of odd order, with nilpotency class 2. For such a group G, we show that if its commutator subgroup coincides with its center, then there exists no non-trivial finite-dimensional pointed Hopf algebra with kG as its coradical. We apply these results to non-abelian groups of order p3, p4 and p5, and list all the pointed Hopf algebras of order p6, whose group of grouplikes is non-abelian.

Actions on p-groups with the kernel containing the 𝔉-residuals

ABSTRACTLet A be a finite group of odd order and let A act on a finite p-group P with |P|>pe, where e is an integer e≥4(e≥5 if p = 2). In this paper we show that P is centralized by if every non-meta-cyclic subgroup of order pe in P is stabilized by Op(A). As applications, some conditions are given for a finite group G with the p-length ≤1 and the p-rank ≤2. We also find a class of finite p-groups, which is not only very useful for the paper but also has its independent meaning.

On the dynamics of endomorphisms of finite groups

Aiming at a better understanding of finite groups as finite dynamical systems, we show that by a version of Fitting’s Lemma for groups, each state space of an endomorphism of a finite group is a graph tensor product of a finite directed 1-tree whose cycle is a loop with a disjoint union of cycles, generalizing results of Hernández-Toledo on linear finite dynamical systems, and we fully characterize the possible forms of state spaces of nilpotent endomorphisms via their “ramification behavior”. Finally, as an application, we will count the isomorphism types of state spaces of endomorphisms of finite cyclic groups in general, extending results of Hernández-Toledo on primary cyclic groups of odd order.

Cayley digraphs of 2-genetic groups of odd prime-power order

A group is called 2-genetic if each normal subgroup of the group can be generated by two elements. Let G be a non-abelian 2-genetic group of order pn for an odd prime p and a positive integer n. In this paper, we investigate connected Cayley digraphs Cay(G,S) for non-abelian 2-genetic groups G of odd order pn, and determine their full automorphism groups A=Aut(Cay(G,S)) in the case when Aut(G,S)={α∈Aut(G)|Sα=S} is a p′-group. It is shown that either Cay(G,S) is normal, that is, the right regular representation of G is normal in A, or p=3,5,7,11 and the largest normal p-subgroup Op(A) of A has order pn−1 with ASL(2,p)≤A/Φ(Op(A))≤AGL(2,p). Furthermore, a non-normal Cayley digraph with smallest order and smallest valency is constructed for each p=3,5,7,11, respectively. In particular, the underlying graphs of the above non-normal Cayley digraphs for p=3,7,11 are half-arc-transitive, and they are the first constructions of half-arc-transitive non-normal Cayley graphs of order a prime-power.

Large $${\varvec{p}}^{\prime }$$ p ′ -orbits for $${\varvec{p}}$$ p -nilpotent linear groups

Let G be a p-nilpotent linear group on a finite vector space V of characteristic p. Suppose that |G||V| is odd. Let P be a Sylow p-subgroup of G. We show that there exist vectors \(v_1\) and \(v_2\) in V such that \(C_G(v_1) \cap C_G(v_2)=P\). A striking conjecture of Malle and Navarro offers a simple global criterion for the nilpotence (in the sense of Broue and Puig) of a p-block of a finite group. Our result implies that this conjecture holds for groups of odd order.

Solvable groups with restrictions on Sylow subgroups of the Fitting subgroup

In this paper, we study solvable groups in which [Formula: see text] is at most 2. In particular, we investigated groups of odd order and [Formula: see text]-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.

Self-Dual Abelian Codes in Some Nonprincipal Ideal Group Algebras

The main focus of this paper is the complete enumeration of self-dual abelian codes in nonprincipal ideal group algebrasF2k[A×Z2×Z2s]with respect to both the Euclidean and Hermitian inner products, wherekandsare positive integers andAis an abelian group of odd order. Based on the well-known characterization of Euclidean and Hermitian self-dual abelian codes, we show that such enumeration can be obtained in terms of a suitable product of the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian self-dual cyclic codes of length2sover some Galois extensions of the ringF2k+uF2k, whereu2=0. Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of lengthpsoverFpk+uFpkare given. Combining these results, the complete enumeration of self-dual abelian codes inF2k[A×Z2×Z2s]is therefore obtained.

On the Galois module structure of the square root of the inverse different in abelian extensions

Let K be a number field with ring of integers OK and let G be a finite group of odd order. Given a G-Galois K-algebra Kh, let Ah denote its square root of the inverse different, which exists by Hilbert's formula. If Kh/K is weakly ramified, then a result of Erez implies that Ah is locally free over OKG and hence defines a class in the locally free class group Cl(OKG) of OKG. In the case that G is abelian, we study the collection of all such classes and show that a subset of them is in fact a subgroup of Cl(OKG).

On Schurity of Finite Abelian Groups

A finite group G is called a Schur group, if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this article, it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any noncyclic abelian Schur group of odd order is isomorphic to ℤ3 × ℤ3 k or ℤ3 × ℤ3 × ℤ p where k ≥ 1 and p is a prime. In addition, we prove that ℤ2 × ℤ2 × ℤ p is a Schur group for every prime p.

Finite group actions on Kervaire manifolds

Let MK4k+2 be the Kervaire manifold: a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product S2k+1×S2k+1 of spheres. We show that a finite group of odd order acts freely on MK4k+2 if and only if it acts freely on S2k+1×S2k+1. If MK is smoothable, then each smooth structure on MK admits a free smooth involution. If k≠2j−1, then MK4k+2 does not admit any free TOP involutions. Free “exotic” (PL) involutions are constructed on MK30, MK62, and MK126. Each smooth structure on MK30 admits a free Z/2×Z/2 action.

Combinatorics of finite abelian groups and Weil representations

The Weil representation of the symplectic group associated to a finite abelian group of odd order is shown to have a multiplicity-free decomposition. When the abelian group is p-primary, the irreducible representations occurring in the Weil representation are parametrized by a partially ordered set which is independent of p. As p varies, the dimension of the irreducible representation corresponding to each parameter is shown to be a polynomial in p which is calculated explicitly. The commuting algebra of the Weil representation has a basis indexed by another partially ordered set which is independent of p. The expansions of the projection operators onto the irreducible invariant subspaces in terms of this basis are calculated. The coefficients are again polynomials in p. These results remain valid in the more general setting of finitely generated torsion modules over a Dedekind domain.

On the Erdős–Ko–Rado property for finite groups

Let a finite group $$G$$G act transitively on a finite set $$X$$X. A subset $$S\subseteq G$$S⊆G is said to be intersecting if for any $$s_1,s_2\in S$$s1,s2?S, the element $$s_1^{-1}s_2$$s1-1s2 has a fixed point. The action is said to have the weak Erd?s---Ko---Rado (EKR) property, if the cardinality of any intersecting set is at most $$|G|/|X|$$|G|/|X|. If, moreover, any maximum intersecting set is a coset of a point stabilizer, the action is said to have the strong EKR property. In this paper, we will investigate the weak and strong EKR property and attempt to classify groups in which all transitive actions have these properties. In particular, we show that a group with the weak EKR property is solvable and that a nilpotent group with the strong EKR property is the direct product of a $$2$$2-group and an abelian group of odd order.

On the existence spectrum for sharply transitive [formula omitted]-designs, [formula omitted] a [formula omitted]-matching

In this paper we consider decompositions of the complete graph Kv into matchings of uniform cardinality k. They can only exist when k is an admissible value, that is a divisor of v(v−1)/2 with 1≤k≤v/2. The decompositions are required to admit an automorphism group Γ acting sharply transitively on the set of vertices. Here Γ is assumed to be either non-cyclic abelian or dihedral and we obtain necessary conditions for the existence of the decomposition when k is an admissible value with 1<k<v/2. Differently from the case where Γ is a cyclic group, these conditions do exclude existence in specific cases. On the other hand we produce several constructions for a wide range of admissible values, in particular for every admissible value of k when v is odd and Γ is an arbitrary group of odd order possessing a subgroup of order gcd(k,v).

Answer to a question on $$A$$ A -groups arisen from the study of Steinitz classes

In this short note we answer to a question of group theory from arXiv:0910.5080. In that paper the author describes the set of realizable Steinitz classes for so-called $A'$-groups of odd order, obtained iterating some direct and semidirect products. It is clear from the definition that $A'$-groups are solvable $A$-groups, but the author left as an open question whether the converse is true. In this note we prove the converse when only two prime numbers divide the order of the group, but we show it to be false in general, producing a family of counterexamples which are metabelian and with exactly three primes dividing the order. Steinitz classes which are realizable for such groups in the family are computed and verified to form a group.