A lower bound for the number of odd-degree representations of a finite group

Even degree characters in principal blocks

We characterise finite groups such that for an odd prime $p$ all the irreducible characters in its principal $p$-block have odd degree. We show that this situation does not occur in non-abelian simple groups of order divisible by $p$ unless $p=7$ and the group is $M_{22}$. As a consequence we deduce that if $p\neq 7$ or if $M_{22}$ is not a composition factor of a group $G$, then the condition above is equivalent to $G/O_{p'}(G)$ having odd order.

Restriction of Odd Degree Characters of $\mathfrak{S}_n$

Let $n$ and $k$ be natural numbers such that $2^k < n$. We study the restriction to $\mathfrak{S}_{n-2^k}$ of odd-degree irreducible characters of the symmetric group $\mathfrak{S}_n$. This analysis completes the study begun in [Ayyer A., Prasad A., Spallone S., Sem. Lothar. Combin. 75 (2015), Art. B75g, 13 pages] and recently developed in [Isaacs I.M., Navarro G., Olsson J.B., Tiep P.H., J. Algebra 478 (2017), 271-282].

$p$ 为奇素数时 $p^6$ 阶群的忠实置换表示的最小次数

Minimal degrees for faithful permutation representations of groups of order $p^6$ where $p$ is an odd prime

本文确定了奇素数 $p$ 下所有 $p^6$ 阶群的忠实置换表示的最小次数，并给出了构造方法。

Hopf Galois structures on extensions of degree twice an odd prime square and their associated left braces

We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.

Hall π-子群与π'度数的特征

Hall $π$-subgroups and characters of $π'$-degree

本文探讨了有限群中Hall π-子群与π'度数特征标之间的关系，并将Navarro和Tiep的结果推广到多个奇素数的情况。

Character degree sums and real representations of finite classical groups of odd characteristic

Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is the power of an odd prime, and let $\mathrm{GSp}(2n, \mathbb{F}_q)$ and $\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$ denote the symplectic and orthogonal groups of similitudes over $\mathbb{F}_q$, respectively. We prove that every real-valued irreducible character of $\mathrm{GSp}(2n, \mathbb{F}_q)$ or $\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$ is the character of a real representation, and we find the sum of the dimensions of the real representations of each of these groups. We also show that if $\boldsymbol{G}$ is a classical connected…

A characterization 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 11/75.

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.

Hopf Galois structures on separable field extensions of odd prime power degree

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(\mathcal{H},μ)$, where $\mathcal{H}$ is a finite cocommutative $K$-Hopf algebra and $μ$ a Hopf action. In this paper, we present several results on Hopf Galois structures on odd prime power degree separable field extensions. We prove that if a separable field extension of odd prime power degree has a Hopf Galois structure of cyclic type, then it has no structure of noncyclic type. We determine the number of Hopf Galois structures of cyclic type on a separable field extension of degree $p^n$, $p$ an odd prime, such that the G…

On the intersection density of primitive groups of degree a product of two odd primes

A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(Ω)$ is intersecting if for any $g,h\in \mathcal{F}$ there exists $ω\in Ω$ such that $ω^g = ω^h$. The \emph{intersection density} $ρ(G)$ of $G$ is the maximum of $\left\{ \frac{|\mathcal{F}|}{|G_ω|} \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\}$, where $G_ω$ is the stabilizer of $ω$ in $G$. In this paper, it is proved that if $G$ is an imprimitive group of degree $pq$, where $p$ and $q$ are distinct odd primes, with at least two systems of imprimitivity then $ρ(G) = 1$. Moreover, if $G$ is primitive of de…

Computation of Hopf Galois structures on separable extensions and classification of those for degree twice an odd prime power

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,μ)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $μ$ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree $2p^n$, with $p$ an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the o…

On super-monomial characters and groups having two irreducible monomial character degrees

A character of a group is said to be super-monomial if every primitive character inducing it is linear. It is conjectured by Isaacs that every irreducible character of an odd $M$-group is super-monomial. We show that all non linear irreducible characters of lowest degree of an odd $M$-group is super-monomial and provide cases in which one can guarantee that certain irreducible characters of normal subgroups are super-monomial. Finally, we study groups having two irreducible monomial character degrees.

极大子群的指数几乎为奇数的有限群

Finite groups whose maximal subgroups have almost odd index

本文研究了由群不变量的算术性质决定有限群结构的问题，确定了极大子群的指数不被4整除的有限群的结构，并由此得到了一些新的2-幂零性准则。

Proportions of elements with given 2-part order in finite classical groups of odd characteristic

For an element $g$ in a group $X$, we say that $g$ has 2-part order $2^{a}$ if $2^{a}$ is the largest power of 2 dividing the order of $g$. We prove lower bounds on the proportion of elements in finite classical groups in odd characteristic that have certain 2-part orders. In particular, we show that the proportion of odd order elements in the symplectic and orthogonal groups is at least $C/\ell^{3/4}$, where $\ell$ is the Lie rank, and $C$ is an explicit constant. We also prove positive constant lower bounds for the proportion of elements of certain 2-part orders independent of the Lie rank. …

Exponents of Zero divisors in the Cohomology ring of a finite group

It is well known that the positive degree cohomology of a finite group G is annihilated by |G|. We improve on this bound in the case of odd degree elements in the integer cohomology ring and show that $e_{odd}(G)$, the exponent of the $\oplus_{k=0}^{\infty} H^{2k+1}(G,\mathbb{Z})$ satisfies $e_{odd}(G)^2$ divides 2|G| and in particular $e_{odd}(G) \leq \sqrt{2|G|}.$ We also provide examples to show this bound for $e_{odd}(G)$ is sharp as a general bound over all finite groups G. The result comes from a fact about zero divisors having "complementary exponent" which we prove using duality in Tat…