Related papers: A Block-theoretic Proof of Burnside's Normal $p$-c…

We show that Thompson's $A\times B$-Lemma can be obtained as a consequence of the Brauer pair version of Brauer's Third Main Theorem.

We prove that for a quasi-regular semiperfectoid $\mathbb{Z}_p^{\rm cycl}$-algebra $R$ (in the sense of Bhatt-Morrow-Scholze), the cyclotomic trace map from the $p$-completed $K$-theory spectrum $K(R;\mathbb{Z}_p)$ of $R$ to the topological…

We use the theory of blocks of cyclic defect to prove that under a certain condition on the principal p-block of a finite group G the normalized unit group of the integral group ring of G contains an element of order pq if and only if so…

In this paper, we list several interesting structures of cyclotomic polynomials: specifically relations among blocks obtained by suitable partition of cyclotomic polynomials. We present explicit and self-contained proof for all of them,…

Let $G$ be a finite group, $p$ a prime number and $P$ a Sylow $p$-subgroup of $G$. Recently, G. Malle, G. Navarro, and P. H. Tiep conjectured that the number of $p$-Brauer characters of $G$ coincides with that of the normaliser ${\bf…

Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr(G|theta) relative to p. We call each member B_theta of this…

In this note we give two proofs of Brooks' Theorem. The first is obtained by modifying an earlier proof and the second by combining two earlier proofs. We believe these proofs are easier to teach in Computer Science courses.

We develop the local-global theory of blocks for profinite groups. Given a field $k$ of characteristic $p$ and a profinite group $G$, one may express the completed group algebra $k[[G]]$ as a product $\prod_{i\in I}B_i$ of closed…

In this note, we present a new proof that the cyclotomic integers constitute the full ring of integers in the cyclotomic field.

Slattery has generalized Brauer's theory of p-blocks of finite groups to pi-blocks of pi-separable groups where pi is a set of primes. In this setting we show that the order of a defect group of a pi-block B is bounded in terms of the…

A well-known theorem of Burnside says that if $\rho$ is a faithful representation of a finite group $G$ over a field of characteristic $0$, then every irreducible representation of $G$ appears as a constituent of a tensor power of $\rho$.…

In this paper, we study some variations of the well-known It\^{o}-Michler theorem for $p$-Brauer characters using various inequalities involving the $p$-Brauer character degrees of finite groups. Several new criteria for the existence of a…

For any prime p and any positive integer n, let B(p,n) denote the n-th layer of the cyclotomic Z_p-extension over the rationals. Based on the heuristics of Cohen and Lenstra, and refined by new results on class numbers of particular B(p,n),…

In this paper, we prove that a \(p\)-block with abelian defect group is inertial if it covers a \(p\)-block of a normal subgroup of \(p\)-power index having only one irreducible Brauer character orbit.

In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…

Borwein and Choi conjectured that a polynomial $P(x)$ with coefficients $\pm1$ of degree $N-1$ is cyclotomic iff $$P(x)=\pm \Phi_{p_1}(\pm x)\Phi_{p_2}(\pm x^{p_1})\cdots \Phi_{p_r}(\pm x^{p_1p_2\cdots p_{r-1}})$$ where $N=p_1p_2\cdots…

We give a probabilistic proof of the orbit-counting lemma.

Let $G$ be a finite group, $N$ a normal subgroup of $G$, and $k$ a field of characteristic $p>0$. In this paper, we formulate the brick version of Clifford's theorem under suitable assumptions and prove it by using the theory of wide…

We prove structural theorems for computing the completion of a G-spectrum at the augmentation ideal of the Burnside ring of a finite group G. First we show that a G-spectrum can be replaced by a spectrum obtained by allowing only isotropy…

In this paper, we give a necessary and sufficient condition for a cyclotomic Brauer algebra being semisimple. This generalizes previous result for a Brauer algebra.