Finite Group Of Odd Order Research Articles

Lee metrics on groups

In this work we consider interval metrics on groups; that is, integral invariant metrics whose associated weight functions do not have gaps. We give conditions for a group to have and not to have interval metrics. Then we study Lee metrics on general groups, that is interval metrics having the finest unitary symmetric associated partition. These metrics generalize the classic Lee metric on cyclic groups. In the case that $ G $ is a torsion-free group or a finite group of odd order, we prove that $ G $ has a Lee metric if and only if $ G $ is cyclic. Also, if $ G $ is a group admitting Lee metrics then $ G \times \mathbb{Z}_2^k $ always has Lee metrics for every $ k \in \mathbb{N} $. Then, we show that some families of metacyclic groups, such as cyclic, dihedral, and dicyclic groups, always have Lee metrics. Finally, we give conditions for non-cyclic groups such that they do not have Lee metrics. We end with tables of all groups of order $ \le 31 $ indicating which of them have (or have not) Lee metrics and why (not).

On the rank of a finite group of odd order with an involutory automorphism

Let G be a finite group of odd order admitting an involutory automorphism $$\phi $$ , and let $$G_{-\phi }$$ be the set of elements of G transformed by $$\phi $$ into their inverses. Note that $$[G,\phi ]$$ is precisely the subgroup generated by $$G_{-\phi }$$ . Suppose that each subgroup generated by a subset of $$G_{-\phi }$$ can be generated by at most r elements. We show that the rank of $$[G,\phi ]$$ is r-bounded.

Primitive characters of odd order groups

Let G be a finite group of odd order. We show that if χ is an irreducible primitive character of G then for all primes p dividing the order of G there is a conjugacy class such that the p-part of χ(1) divides the size of that conjugacy class. We also show that for some classes of groups the entire degree of an irreducible primitive character χ divides the size of a conjugacy class.

Exponent of a finite group of odd order with an involutory automorphism

Let $G$ be a finite group of odd order admitting an involutory automorphism $\phi$. We obtain two results bounding the exponent of $[G,\phi]$. Denote by $G_{-\phi}$ the set $\{[g,\phi]\,\vert\, g\in G\}$ and by $G_{\phi}$ the centralizer of $\phi$, that is, the subgroup of fixed points of $\phi$. The obtained results are as follows.1. Assume that the subgroup $\langle x,y\rangle$ has derived length at most $d$ and $x^e=1$ for every $x,y\in G_{-\phi}$. Suppose that $G_\phi$ is nilpotent of class $c$. Then the exponent of $[G,\phi]$ is $(c,d,e)$-bounded.2. Assume that $G_\phi$ has rank $r$ and $x^e=1$ for each $x\in G_{-\phi}$. Then the exponent of $[G,\phi]$ is $(e,r)$-bounded.

Iteration of the exterior power on representation rings

In this note we investigate the idea of Michael Atiyah of using, as a possible approach to the Theorem of Feit–Thompson on the solvability of finite groups of odd order, the iterations of the transformation which replaces a representation of a finite group G on a finite dimensional complex vector space E by the difference between the associated representation of G on the sum of exterior powers of E and the trivial representation. We show that G has odd order if and only if the above operation extends to virtual representations and we then express it in terms of the exponential and the Adams operations in the complexified representation ring. We show the relevance of the idea in a concrete example by exhibiting convergence to a non-trivial character.

Three questions of Bertram on locally maximal sum-free sets

Let G be a finite group, and S a sum-free subset of G. The set S is locally maximal in G if S is not properly contained in any other sum-free set in G. If S is a locally maximal sum-free set in a finite abelian group G, then $$G=S\cup SS\cup SS^{-1}\cup \sqrt{S}$$ , where $$SS=\{xy|~x,y\in S\}$$ , $$SS^{-1}=\{xy^{-1}|~x,y\in S\}$$ and $$\sqrt{S}=\{x\in G|~x^2\in S\}$$ . Each set S in a finite group of odd order satisfies $$|\sqrt{S}|=|S|$$ . No such result is known for finite abelian groups of even order in general. In view to understanding locally maximal sum-free sets, Bertram asked the following questions: In this paper, we answer question (i) in the negative, then (ii) and (iii) in affirmative.

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 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).

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.

Odd-order Cayley graphs with commutator subgroup of order pq are hamiltonian

We show that if G is a nontrivial, finite group of odd order, whose commutator subgroup [ G , G ] is cyclic of order p μ q ν , where p and q are prime, then every connected Cayley graph on G has a hamiltonian cycle.

On 10-Centralizer Groups of Odd Order

Let G be a group, and let Cent(G) denote the number of distinct centralizers of its elements. A group G is called n-centralizer if Cent(G)=n. In this paper, we investigate the structure of finite groups of odd order with Cent(G)=10 and prove that there is no finite nonabelian group of odd order with Cent(G)=10.

Splitting fields of finite groups

We give a simpler proof of the Goldschmidt-Isaacs theorem in the case and find new sufficient conditions for the applicability of the theorem in the case . We thus obtain a theorem giving an estimate for the Schur index of an arbitrary irreducible complex representation of a finite group over the field of rational numbers. The proof of this theorem shows that in practical applications there is no need to verify sufficient conditions for the applicability of the Goldschmidt-Isaacs theorem in the case : they can automatically be assumed to hold. We also prove a theorem on the connection between the realizability of any complex representation over the field of rational numbers of a finite group of odd order of a special type and the possibility of constructing regular polygons with straightedge and compasses.

On Groups of Odd Order Admitting an Elementary 2-Group of Automorphisms

Let G be a finite group of odd order with derived length k . We show that if G is acted on by an elementary abelian group A of order 2^n and C_G(A) has exponent e , then G has a normal series G=G_0\ge T_0\ge G_1\ge T_1\ge\cdots\ge G_n\ge T_n=1 such that the quotients G_i/T_i have \{k,e,n\} -bounded exponent and the quotients T_i/G_{i+1} are nilpotent of \{k,e,n\} -bounded class.

On the existence of self-dual permutation codes of finite groups

Motivated by a research on self-dual extended group codes, we consider permutation codes obtained from submodules of a permutation module of a finite group of odd order over a finite field, and demonstrate that the condition “the extension degree of the finite field extended by n’th roots of unity is odd” is sufficient but not necessary for the existence of self-dual extended transitive permutation codes of length n + 1. It exhibits that the permutation code is a proper generalization of the group code, and has more delicate structure than the group code.

A CHARACTERISATION OF CERTAIN FINITE GROUPS OF ODD ORDER

The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The main object of this paper is to obtain a characterization for all finite groups of odd order with commutativity degree greater than or equal to 11 .

A CHARACTERISATION OF CERTAIN FINITE GROUPS OF ODD ORDER

The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The main object of this paper is to obtain a characterisation for all finite groups of odd order with commutativity degree greater than or equal to $\frac{11}{75}$ .

Self-dual modules for finite groups of odd order

Let G be a finite group of odd order and let F be a finite field. Suppose that V is an FG-module which carries a G-invariant non-degenerate bilinear form which is symmetric or symplectic. We show that V contains a self-perpendicular submodule if and only if the characteristic polynomials of some specified elements of G (regarded as linear transformations of V) are precisely squares. This result can be applied to the study of monomial characters if the form on V is symplectic, and self-dual group codes if the form is symmetric.

Quadratic characters in groups of odd order

We prove that in a finite group of odd order, the number of irreducible quadratic characters is the number of quadratic conjugacy classes.

Centralizers of involutory automorphisms of groups of odd order

The following theorem is proved. Let G be a finite group of odd order admitting an involutory automorphism ϕ such that G = [ G , ϕ ] . Suppose that C G ( ϕ ) has a nilpotent subgroup of index n. Then the index [ G ′ : F ( G ′ ) ] is bounded by a function depending only on n.

On groups of odd order with exactly two non-central conjugacy classes of each size

The non-abelian group of order 21 is the only finite group of odd order with exactly two non-central conjugacy classes of each size.