$p$-groups and zeros of characters
Fix a prime $p$ and an integer $n\geq 0$. Among the non-linear irreducible characters of the $p$-groups of order $p^n$, what is the minimum number of elements that take the value 0?
math.GR↗
arXiv subjects
Gabriel Navarro
Publications and source records attributed to Gabriel Navarro.
The Field of Values of the Height Zero Characters
We determine what are the fields of values of the irreducible $p$-height zero characters of all finite groups for $p=2$; we conjecture what they should be for odd primes, and reduce this statement to a problem on blocks of quasi-simple groups.
math.GR↗
Characters, Commutators and Centers of Sylow Subgroups
The character table of a finite group G determines whether |P:P'|=p^2 and whether |P:Z(P)|=p^2, where P is a Sylow p-subgroup of G. To prove the latter, we give a detailed classification of those groups in terms of the generalized Fitting subgroup.
math.RT↗
Height Zero Conjecture with Galois Automorphisms
We prove a strengthening of Brauer's height zero conjecture for principal 2-blocks with Galois automorphisms. This requires a new extension of the Itô--Michler theorem for the prime~2, again with Galois automorphisms. We close, this time for odd primes $p$, with a new characterisation of $p$-closed groups via the decomposition numbers of certain characters.
math.RT↗
Skew differential Goppa codes and their application to McEliece cryptosystem
A class of linear codes that extends classic Goppa codes to a non-commutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these codes, when the alphabet is a finite field, may be adjusted to propose a McEliece-type cryptosystem.
cs.IT↗
On the commuting probability of p-elements in a finite group
Let $G$ be a finite group, let $p$ be a prime and let ${\rm Pr}_p(G)$ be the probability that two random $p$-elements of $G$ commute. In this paper we prove that ${\rm Pr}_p(G) > (p^2+p-1)/p^3$ if and only if $G$ has a normal and abelian Sylow $p$-subgroup, which generalizes previous results on the widely studied commuting probability of a finite group. This bound is best possible in the sense that for each prime $p$ there are groups with ${\rm Pr}_p(G) = (p^2+p-1)/p^3$ and we classify all such groups. Our proof is based on bounding the proportion of $p$-elements in $G$ that commute with a fixed $p$-element in $G \setminus \textbf{O}_p(G)$, which in turn relies on recent work of the first two authors on fixed point ratios for finite primitive permutation groups.
math.GR↗
Principal blocks for different primes, II
If G is a finite group, we have proposed new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper, we give strong support to the validity of these conjectures.
math.GR↗
Principal blocks for different primes, I
We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures.
math.GR↗
Alternating sums over pi-subgroups
Dade's conjecture predicts that if p is a prime, then the number of irreducible characters of a finite group of a given p-defect is determined by local subgroups. In this paper we replace $p$ by a set of primes pi and prove a pi-version of Dade's conjecture for pi-separable groups. This extends the (known) p-solvable case of the original conjecture and relates to a pi-version of Alperin's weight conjecture previously established by the authors.
math.RT↗
Brauer's Height Zero Conjecture for Principal Blocks
We prove \emph{the other half} of Brauer's Height Zero Conjecture in the case of principal blocks.
math.RT↗
Real Constituents of Permutation Characters
We prove a broad generalization of a theorem of W. Burnside on real characters using permutation characters. Under a necessary hypothesis, We can give some control on multiplicities (a result that needs the Classification of Finite Simple Groups). Along the way, we also give a new characterization of the 2-closed finite groups using odd-order real elements of the group. All this can be seen as a contribution to Brauer's Problem 11. We also obtain similar results for 2-Brauer characters. We also classify finite primitive permutation groups in which every real element has a fixed point.
math.GR↗
Peterson-Gorenstein-Zierler algorithm for differential convolutional codes
Differential Convolutional Codes with designed Hamming distance are defined, and an algebraic decoding algorithm, inspired by Peterson-Gorenstein-Zierler's algorithm, is designed for them.
cs.IT↗
Decoding Reed-Solomon Skew-Differential Codes
A large class of MDS linear codes is constructed. These codes are endowed with an efficient decoding algorithm. Both the definition of the codes and the design of their decoding algorithm only require from Linear Algebra methods, making them fully accesible for everyone. Thus, the first part of the paper develops a direct presentation of the codes by means of parity-check matrices, and the decoding algorithm rests upon matrix and linear maps manipulations. The somewhat more sophisticated mathematical context (non-commutative rings) needed for the proof of the correctness of the decoding algorithm is postponed to the second part. A final section locates the Reed-Solomon skew-differential codes introduced here within the general context of codes defined by means of skew polynomial rings.
cs.IT↗
Biseparable extensions are not necessarily Frobenius
We give necessary and sufficient conditions on an Ore extension $A[x;σ,δ]$, where $A$ is a finite dimensional algebra over a field $\mathbb{F}$, for being a Frobenius extension over the ring of commutative polynomials $\mathbb{F}[x]$. As a consequence, as the title of this paper highlights, we provide a negative answer to a problem stated by Caenepeel and Kadison.
math.RA↗
Some remarks on non projective Frobenius algebras and linear codes
With a small suitable modification, dropping the projectivity condition, we extend the notion of a Frobenius algebra to grant that a Frobenius algebra over a Frobenius commutative ring is itself a Frobenius ring. The modification introduced here also allows Frobenius finite rings to be precisely those rings which are Frobenius finite algebras over their characteristic subrings. From the perspective of linear codes, our work expands one's options to construct new finite Frobenius rings from old ones. We close with a discussion of generalized versions of the MacWilliams identities that may be obtained in this context.
math.RA↗
A reduction theorem for the Galois-McKay conjecture
We generalize the theory of ordering character triples, developed by Navarro and Späth, by taking into account the action of Galois automorphisms on characters. This new technique, together with previous results of Ladisch and Turull, allows us to reduce the Galois--McKay conjecture to a question about simple groups.
math.RT↗
Centers of Sylow Subgroups and Automorphisms
Suppose that p is an odd prime and G is a finite group having no normal non-trivial p'-subgroup. We show that if a is an automorphism of G of p-power order centralizing a Sylow p-group of G, then a is inner. This answers a conjecture of Gross. An easy corollary is that if p is an odd prime and P is a Sylow p-subgroup of G, then the center of P is contained in the generalized Fitting subgroup of G. We give two proofs both requiring the classification of finite simple groups. For p=2, the result fails but Glauberman in 1968 proved that the square of a is inner. This answered a problem of Kourovka posed in 1999.
math.GR↗
Irreducible induction and nilpotent subgroups in finite groups
Suppose that $G$ is a finite group and $H$ is a nilpotent subgroup of $G$. If a character of $H$ induces an irreducible character of $G$, then the generalized Fitting subgroup of $G$ is nilpotent.
math.RT↗