جستجوی مقالات مرتبط با کلیدواژه « character degree » در نشریات گروه « ریاضی »

Denote by $\widehat{p_n}$, the largest prime among the primitive prime divisors of $ 2^{2n+1}-1 $ and $ 2^{2(4n+2)}-1 $, where $n\in {\Bbb N}$. In this paper, we prove that if $ q=2^{2n+1}\geq8 $ and $\alpha \leq \widehat{p_n}$, then the direct product of $ \alpha $ copies of $ {\rm Sz}(q)$ is uniquely determined by its complex group algebra.

Let G be a finite group and cd∗ (G) be the set of nonlinear irreducible character degrees of G. Suppose that ρ(G) denotes the set of primes dividing some element of cd∗ (G). The bipartite divisor graph for the set of character degrees which is denoted by B(G), is a bipartite graph whose vertices are the disjoint union of ρ(G) and cd∗ (G), and a vertex p ∈ ρ(G) is connected to a vertex a ∈ cd∗ (G) if and only if p|a. In this paper, we investigate the structure of a group G whose graph B(G) has five vertices. Especially we show that all these groups are solvable.

Let $G$ be a finite group and $cd(G)$ denote the character degree set for $G$. The prime graph $DG$ is a simple graph whose vertex set consists of prime divisors of elements in $cd(G)$, denoted $rho(G)$. Two primes $p,qin rho(G)$ are adjacent in $DG$ if and only if $pq|a$ for some $ain cd(G)$. We determine which simple 4-regular graphs occur as prime graphs for some finite nonsolvable group.

In a previous paper, the second author established that, given finite fields $F < E$ and certain subgroups $C leq E^times$, there is a Galois connection between the intermediate field lattice ${L mid F leq L leq E}$ and $C$''s subgroup lattice. Based on the Galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product $C rtimes {Gal} (E/F)$. However, the analysis when $|F|$ is a Mersenne prime is more complicated, so certain cases were omitted from that paper.The present exposition, which is a reworking of the previous article, provides a uniform analysis over all the families, including the previously undetermined ones. In the group $Crtimes{rm Gal(E/F)}$, we use the Galois connection to calculate stabilizers of linear characters, and these stabilizers determine the full character degree set. This is shown for each subgroup $Cleq E^times$ which satisfies the condition that every prime dividing $|E^times: C|$ divides $|F^times|$.