Generalized circulants and class functions of finite groups. II

AbstractIn a previous article it was shown that the semisimple algebra of class functions of a finite group gives rise to a family of matrices that enjoys many of the properties of circulant matrices. Varying the groups yields different families (circulants, block circulants of all levels, and others), whose properties are simple consequences of character theory. In this article we observe that many more such families can be constructed by a slight modification of the method used in the first paper. The new set of families includes all the families obtained in the first paper, the families of skew-circulants, {k}-circulants, retrocirculants, some g-circulants, and many others. Known properties of these specific families are special cases of the properties of the general families constructed, many of which are themselves corollaries of simple facts about the regular representation of semisimple commutative algebras

Finite groups with almost distinct character degrees

AbstractFinite groups with the nonlinear irreducible characters of distinct degrees, were classified by the authors and Berkovich. These groups are clearly of even order. In groups of odd order, every irreducible character degree occurs at least twice. In this article we classify finite nonperfect groups G, such that χ(1)=θ(1) if and only if θ=χ¯ for any nonlinear χ≠θ∈Irr(G). We also present a description of finite groups in which xG′⊆class(x)∪class(x−1) for every x∈G−G′. These groups generalize the Frobenius groups with an abelian complement, and their description is needed for the proof of the above mentioned result on characters

On a problem of Frobenius

AbstractLet G be a finite group and let n be a natural integer. We define nG = (n, |G|) and Ln(G) = {g ϵ G| gn = 1}. We shall write G ϵ Fn if |Ln(G)| = nG. Frobenius conjectured that if G ϵ Fn, then Ln(G) is a normal subgroup of G (or, in short, G is n-closed). A weaker conjecture of Frobenius states that if H ϵ Fn for every subgroup H of a finite group G (including G itself), then G is n-closed. This weaker conjecture is proved in this article for natural numbers n not divisible by 4. In fact, our result is more general than the weaker Frobenius conjecture

Lockable Knee Implants and Related Methods

Total knee replacements for hinged knee implants include a tibial member, a femoral member, a hinge assembly having a laterally extending axle configured to hingedly attach the femoral member to the tibial member, and a lock mechanism in communication with the hinge assembly. The lock mechanism is configured to (i) lock the femoral member in alignment with the tibial member for a full extension or other defined stabile walking configuration to thereby allow an arthrodesis or stiff knee gait and (ii) unlock to allow the femoral and tibial members to pivot relative to each other for flexion or bending when not ambulating

Lockable Knee Implants and Related Methods

Total knee replacements for hinged knee implants include a tibial member, a femoral member, a hinge assembly having a laterally extending axle configured to hingedly attach the femoral member to the tibial member, and a lock mechanism in communication with the hinge assembly. The lock mechanism is configured to (i) lock the femoral member in alignment with the tibial member for a full extension or other defined stabile walking configuration to thereby allow an arthrodesis or stiff knee gait and (ii) unlock to allow the femoral and tibial members to pivot relative to each other for flexion or bending when not ambulating

Lockable Implants

Total joint replacements for implants include a first member configured to attach to a first bone, a second member configured to reside in an adjacent second bone and a locking mechanism. The locking mechanism is configured to (i) lock the first and second members in alignment for full extension or other defined stabilized configuration and (ii) unlock to allow the first and second members to pivot relative to each other for flexion or bending

Semisimple commutative algebras with positive bases

Algebras that serve as models for concurrent studying of certain aspects of both the algebra of ordinary characters and the center of the group algebra have been considered by various authors. In this article we o¤er another such model. The main di¤erences between our model and the known ones are: 1. Our model includes Brauer characters and principal indecomposable characters as special cases. 2. Our emphasis is on the eigenvalues of the regular representation of the algebra elements, an approach that gives results on values of characters (ordinary, central, Brauer, principal indecomposable) as well as on their products. Supported by the fund for the promotion of research at the Technion.

Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes

AbstractLet A be a semisimple, n-dimensional, commutative algebra over a field F. Fix a basis B of A, and denote by M(a; B) the transpose of the matrix over F that represents a ϵ A regularly with respect to B. It is easy to see that the set {M(a; B) |a ϵ A} can be simultaneously diagonalized over many fields (including all perfect fields). We use this fact in order to give an elementary proof that such an algebra over an infinite field is generated by a single element, and to describe the subalgebras of A in terms of certain partitions of the set {1,2,3,…, n}. Several applications of these results are shown: (1) We give a new proof for the theorem stating that every finite-dimensional, separable field extension has a primitive element. (2) We show that every finite group G has a character θ such that every other generalized character of G is a polynomial in θ with rational coefficients. (This is true for Brauer characters as well.) (3) We give a necessary condition for two generalized characters (or Brauer characters) ζ and χ that forces the field of values of ζ to contain that of χ. (4) Many collections of patterned matrices over a field F, such as circulant matrices and some of their generalizations are known to be algebras generated by a single matrix. We observe that each subalgebra of such a collection is also generated by a single matrix. Also, if a and b are two elements of such a collection, we give a necessary and sufficient condition, in terms of the eigenvalue pattern of a and b, for a to be a polynomial in b with coefficients in F. (5) We show that if A is a (generalized) cyclic code, then the eigenvalues of M(a; B) are the so-called Matteson-Solomon coefficients of the codeword a. Other applications to coding, to groups, and to field extensions are discussed as well

ON TRANSITIVE PERMUTATION GROUPS WITH AN IMPRIMITIVITY BLOCK OF PRIME LENGTH

ABSTRACT. Let G be a transitive permutation group on a finite set 12. Let 23 ={A 1,A2,...,At) be a system of imprimitivity for G on 12 with [All = p, a prime. Assume that the kernel of G on 23 is not faithful on A 1. It is proved that either G contains a nontrivial normal elementary abelian p-subgroup or the minimal degree of G is less than-•-J-2 ' Smaller bounds for the minimal degree are found in special cases and some corollaries on multiply transitive groups are obtained. I. Introduction. Let G be a finite group and H and K subgroups of G such that 1 4: H C K 4: G and [K:HI = p, a prime. If G = G/CqxC__GHX and K and H are the images-- _ of K and H, then G satisfies the assumption of the title in its action on the cosets of H (see [6], page 15). Thus "almost " every finite group has a homomorphic imag