Groups whose vanishing class sizes are prime powers

For a finite group 𝐺, an element is called a vanishing element of 𝐺 if it is a zero of an irreducible character of 𝐺; otherwise, it is called a non-vanishing element. Moreover, the conjugacy class of an element is called a vanishing class if that element is a vanishing element. In this paper, we describe finite groups whose vanishing class sizes are all prime powers, and on the other hand we show that non-vanishing elements of such a group lie in the Fitting subgroup which is a proof of a conjecture mentioned in [I. M. Isaacs, G. Navarro and T. R. Wolf, Finite group elements where no irreducible character vanishes, J. Algebra 222 (1999), 2, 413–423] under this special restriction on vanishing class sizes.

On Groups Whose Irreducible Character Degrees of All Proper Subgroups are All Prime Powers

Isaacs, Passman, and Manz have determined the structure of finite groups whose each degree of the irreducible characters is a prime power. In particular, if G is a nonsolvable group and every character degree of a group G is a prime power, then G is isomorphic to S × A , where S ∈ A 5 , PSL 2 8 and A is abelian. In this paper, we change the condition, each character degree of a group G is a prime power, into the condition, each character degree of the proper subgroups of a group is a prime power, and give the structure of almost simple groups whose character degrees of all proper subgroups are all prime powers.

NONDIVISIBILITY AMONG IRREDUCIBLE CHARACTER CO-DEGREES

For a character $\chi $ of a finite group G, the number $\chi ^c(1)={[G:{\textrm {ker}}\chi ]}/{\chi (1)}$ is called the co-degree of $\chi $ . A finite group G is an ${\textrm {NDAC}} $ -group (no divisibility among co-degrees) when $\chi ^c(1) \nmid \phi ^c(1)$ for all irreducible characters $\chi $ and $\phi $ of G with $1< \chi ^c(1) < \phi ^c(1)$ . We study finite groups admitting an irreducible character whose co-degree is a given prime p and finite nonsolvable ${\textrm {NDAC}} $ -groups. Then we show that the finite simple groups $^2B_2(2^{2f+1})$ , where $f\geq 1$ , $\mbox {PSL}_3(4)$ , ${\textrm {Alt}}_7$ and $J_1$ are determined uniquely by the set of their irreducible character co-degrees.

Algorithmically Distinguishing Irreducible Characters of the Symmetric Group

Suppose that $\chi_\lambda$ and $\chi_\mu$ are distinct irreducible characters of the symmetric group $S_n$. We give an algorithm that, in time polynomial in $n$, constructs $\pi\in S_n$ such that $\chi_\lambda(\pi)$ is provably different from $\chi_\mu(\pi)$. In fact, we show a little more. Suppose $f = \chi_\lambda$ for some irreducible character $\chi_\lambda$ of $S_n$, but we do not know $\lambda$, and we are given only oracle access to $f$. We give an algorithm that determines $\lambda$, using a number of queries to $f$ that is polynomial in $n$. Each query can be computed in time polynomial in $n$ by someone who knows $\lambda$.

Finite groups with at most six vanishing conjugacy classes

Let [Formula: see text] be a finite group. We say that a conjugacy class of [Formula: see text] in [Formula: see text] is vanishing if there exists some irreducible character [Formula: see text] of [Formula: see text] such that [Formula: see text]. In this paper, we show that finite groups with at most six vanishing conjugacy classes are solvable or almost simple groups.

Carnival as Contentious Performance

In the 1970s, in a context of increased racial tensions and growing nationalist claims, the use of rhythms, instruments, and clothing associated with Africa among the black populations of England, Guadeloupe, and Martinique became part of a cultural and political repertoire aimed at resurrecting and denouncing a long history of subordination. Similarly, the mobilization of carnival by Afro-Caribbean activists today can be considered as a tactical choice—that is to say, carnival has become part of the standardized, limited, context-dependent repertoires from which claim-making performances are drawn. Based on ethnographic fieldwork conducted in Fort-de-France, London, and Pointe-à-Pitre between 2000 and 2018, this article analyzes how cultural movements have drawn on carnivalesque aesthetics to both memorialize and display the complex history of black Caribbean populations. I argue that Caribbean carnival has been subject to constant reinterpretations since the eighteenth century and that, as such, this repertoire is not only a model or a set of limited means of action, but also a convention through which carnival groups constantly reinvent their skills and resources. Furthermore, this article shows that the repertoires mobilized by the carnival bands I study in Europe and in the Caribbean cannot be reduced to an aesthetic gesture that serves political claims, and that they are part of a historical genealogy that testifies to the irreducible character of a way of life.

𝒫-characters and the structure of finite solvable groups

Let 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.