Let B be a block of a finite group G with defect group D. We prove that the exponent of the center of D is determined by the character table of G. In particular, we show that D is cyclic if and only ...if B contains a “large” family of irreducible p-conjugate characters. More generally, for abelian D we obtain an explicit formula for the exponent of D in terms of character values. In small cases even the isomorphism type of D is determined in this situation. Moreover, it can read off from the character table whether |D/D′|=4 where D′ denotes the commutator subgroup of D. We also propose a new characterization of nilpotent blocks in terms of the character table.
•A unique symmetry-adapted convention for nodes and nodal freedoms is defined.•Automatic generation of the group-theoretic transformation matrix is explained.•A consistent procedure for ...block-diagonalization of system matrix is developed.•Validity of method is demonstrated by consideration of the vibration of a grid.
Symmetry properties of physical systems may be studied through symmetry groups. In recent times, group theory has found application in the study of various problems in structural mechanics, specifically bifurcation, buckling, kinematics and vibration. Computational simplifications are achieved by decomposing the vector space of the problem into smaller subspaces that are independent of each other. When the basis vectors of a subspace are used as the symmetry-adapted variables of that subspace, a smaller problem (associated with a matrix of smaller dimensions) automatically results. However, the same decomposition may be achieved by first obtaining the structural matrix of the system, and then transforming this into a non-overlapping block-diagonal matrix, each independent block being associated with a subspace of the problem. The advantage of this approach is its greater amenability to computer programming, but it does not always give the correct results unless a very specific procedure is followed. The purpose of this contribution is to present a consistent group-theoretic approach for the block diagonalization of structural matrices.
We determine character tables for twisted fractional linear groups that form the "other" family in Zassenhaus' classification of finite sharply 3-transitive groups.
Character table sudokus Sambale, Benjamin
Archiv der Mathematik,
07/2023, Letnik:
121, Številka:
1
Journal Article
Recenzirano
Odprti dostop
It is a fun game to complete a partial character table of a finite group. We show that one can reconstruct a missing row or column from a given table. The proof relies on deep properties of fully ...ramified characters. Moreover, we extend a classification of groups with a “large” character degree started by Snyder and continued by Durfee and Jensen.
The nth cyclotomic polynomial Φn(X) is the minimal polynomial of ζn:=e2πi/n. Given an integer m≥1 and a prescribed set S of arithmetic progressions modulo m, we define nx as the product of the primes ...p≤x lying in those progressions. Let d(n) denote the number of divisors of n. It turns out that under certain conditions on S and m there exists jx such that log|Φnx(ζmjx)|/d(nx) tends to a positive limit. Our aim is to determine those conditions. We use the arithmetic of cyclotomic number fields, non-standard properties of character tables of finite abelian groups and a recent theorem of Bzdȩga, Herrera-Poyatos and Moree. After developing some generalities, we restrict to the case where m is a prime.
Our motivation comes from a paper of Vaughan (1975). He studied the case where S={±2(mod5)} and used it to show that the maximum coefficient in absolute value of Φn can be very large.
We determine a supercharacter theory for Sylow p-subgroups G2syl2(32m+1) of the Ree groups G22(32m+1), calculate the conjugacy classes of G2syl2(32m+1), and establish the character table of G2syl2(3).
Abstract
Xu characterized rank 4 self‐dual association schemes inducing three partial geometric designs by their character tables. We construct such association schemes as Schur rings over Abelian ...2‐groups.
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