Fusion in the character table

Suppose that P P is a Sylow p p -subgroup of a finite p p -solvable group G G . If g ∈ P g \in P , then the number of G G -conjugates of g g in P P can be read off from the character table of G G .

On the number of covering blocks

A counterexample to Feit's Problem VIII on decomposition numbers

We find a counterexample to Feit's Problem VIII on the bound of decomposition numbers. This also answers a question raised by T. Holm and W. Willems.

The minimal number of characters over a normal p-subgroup

Abstract If N is a normal p-subgroup of a finite group G and θ ∈ Irr ( N ) is a G-invariant irreducible character of N, then the number | Irr ( G | θ ) | of irreducible characters of G over θ is always greater than or equal to the number k p ′ ( G / N ) of conjugacy classes of G / N consisting of p ′ -elements. In this paper, we investigate when there is equality.

Sylow normalizers and character tables, II

Suppose thatG is a finitep-solvable group and letPe Syl p (G). In this note, we prove that the character table ofG determines ifN G(itP)/P is abelian.

Character restrictions and multiplicities in symmetric groups

Abstract We give natural correspondences of odd-degree characters of the symmetric groups and some of their subgroups, which can be described easily by restriction of characters, degrees and multiplicities.

The First Main Theorem

Characters of relative p'-degree over normal subgroups

Let Z be a normal subgroup of a finite group G , let ??Irr(Z) be an irreducible complex character of Z , and let p be a prime number. If p does not divide the integers ?(1)/?(1) for all ??Irr(G) lying over ? , then we prove that the Sylow p -subgroups of G/Z are abelian. This theorem, which generalizes the Gluck-Wolf Theorem to arbitrary finite groups, is one of the principal obstacles to proving the celebrated Brauer Height Zero Conjecture

Two groups with isomorphic group algebras

Characters, bilinear forms and solvable groups

Abstract We prove a number of results about the ordinary and Brauer characters of finite solvable groups in characteristic 2, by defining and using the concept of the extended nucleus of a real irreducible character. In particular we show that the Isaacs canonical lift of a real irreducible Brauer character has Frobenius–Schur indicator +1. We also show that the principal indecomposable module corresponding to a real irreducible Brauer character affords a quadratic geometry if and only if each extended nucleus is a split extension of a nucleus.

Brauer characters with cyclotomic field of values

It has been shown in an earlier paper [G. Navarro, Pham Huu Tiep, Rational Brauer characters, Math. Ann. 335 (2006) 675–686] that, for any odd prime p, every finite group of even order has a non-trivial rational-valued irreducible p-Brauer character. For p=2 this statement is no longer true. In this paper we determine the possible non-abelian composition factors of finite groups without non-trivial rational-valued irreducible 2-Brauer characters. We also prove that, if p≠q are primes, then any finite group of order divisible by q has a non-trivial irreducible p-Brauer character with values in the cyclotomic field Q(exp(2πi/q)).

Brauer Characters Relative to a Normal Subgroup

Brauer’s Height Zero Conjecture for principal blocks

Abstract We prove the other half of Brauer’s Height Zero Conjecture in the case of principal blocks.

Non-vanishing elements of finite groups

AbstractLet G be a finite group, and let Irr(G) denote the set of irreducible complex characters of G. An element x of G is non-vanishing if, for every χ in Irr(G), we have χ(x)≠0. We prove that, if x is a non-vanishing element of G and the order of x is coprime to 6, then x lies in the Fitting subgroup of G.

McKay natural correspondences on characters

Let [math] be a finite group, let [math] be an odd prime, and let [math] . If [math] , then there is a canonical correspondence between the irreducible complex characters of [math] of degree not divisible by [math] belonging to the principal block of [math] and the linear characters of [math] . As a consequence, we give a characterization of finite groups that possess a self-normalizing Sylow [math] -subgroup or a [math] -decomposable Sylow normalizer.

Local functions on finite groups

We study local properties of finite groups using chains of p p -subgroups.

Character degrees and local subgroups of 𝜋-separable groups

Let G G be a finite { p , q } \{p,q \} -solvable group for different primes p p and q q . Let P ∈ Syl p ( G ) P \in \text {Syl}_{p}(G) and Q ∈ Syl q ( G ) Q \in \text {Syl}_{q}(G) be such that P Q = Q P PQ=QP . We prove that every χ ∈ Irr ( G ) \chi \in \text {Irr}(G) of p ′ p^{\prime } -degree has q ′ q^{\prime } -degree if and only if N G ( P ) ⊆ N G ( Q ) \mathbf {N}_{G}(P) \subseteq \mathbf {N}_{G}(Q) and C Q ′ ( P ) = 1 \mathbf {C}_{Q^{\prime }}(P)=1 .

The set of conjugacy class sizes of a finite group does not determine its solvability

Abstract We find a pair of groups, one solvable and the other non-solvable, with the same set of conjugacy class sizes.

Characters that agree on prime-power-order elements

Rationality and Sylow 2-subgroups

AbstractLet G be a finite group. If G has a cyclic Sylow 2-subgroup, then G has the same number of irreducible rational-valued characters as of rational conjugacy classes. These numbers need not be the same even if G has Klein Sylow 2-subgroups and a normal 2-complement.

𝑝-rational characters and self-normalizing Sylow 𝑝-subgroups

Let G G be a finite group, p p a prime, and P P a Sylow p p -subgroup of G G . Several recent refinements of the McKay conjecture suggest that there should exist a bijection between the irreducible characters of p ′ p’ -degree of G G and the irreducible characters of p ′ p’ -degree of N G ( P ) \mathbf {N}_G(P) , which preserves field of values of correspondent characters (over the p p -adics). This strengthening of the McKay conjecture has several consequences. In this paper we prove one of these consequences: If p > 2 p>2 , then G G has no non-trivial p ′ p’ -degree p p -rational irreducible characters if and only if N G ( P ) = P \mathbf {N}_G(P)=P .

Characters of p′-Degree of p-Solvable Groups

Character Tables and Sylow Subgroups Revisited

Suppose that G is a finite group. A classical and difficult problem is to determine how much the character table knows about the local structure of G and vice versa.

Restricting irreducible characters to Sylow 𝑝-subgroups

We restrict irreducible characters of finite groups of degree divisible by p p to their Sylow p p -subgroups and study the number of linear constituents.

The Second Main Theorem

Pronormal subgroups and zeros of characters

We give a characterization of when a pronormal subgroup of a solvable group is normal by using character theory.

Zeros of Primitive Characters in Solvable Groups

Real groups and Sylow 2-subgroups

Abstract If G is a finite real group and P ∈ Syl 2 ( G ) , then P / P ′ is elementary abelian. This confirms a conjecture of Roderick Gow. In fact, we prove a much stronger result that implies Gow's conjecture.

On defects of characters and decomposition numbers

We propose upper bounds for the number of modular constituents of the restriction modulo [math] of a complex irreducible character of a finite group, and for its decomposition numbers, in certain cases.

Global–Local Counting Conjectures

A conjecture on the number of conjugacy classes in ap-solvable group

IfG is ap-solvable group, it is conjectured that k(G/O P (G) ≤ |G| p ′. The conjecture is easily obtained for solvable groups as a consequence of R. Knorr’s work on the k(GV) problem. Also, a related result is obtained: k(G/F(G)) is bounded by the index of a nilpotent injector ofG.

Character sums and double cosets

Abstract If G is a p-solvable finite group, P is a self-normalizing Sylow p-subgroup of G with derived subgroup P ′ , and Ψ is the sum of all the irreducible characters of G of degree not divisible by p, then we prove that the integer Ψ ( P ′ z P ′ ) is divisible by | P | for all z ∈ G . This answers a question of J. Alperin.

Characterizing normal Sylow p-subgroups by character degrees

Abstract Suppose that G is a finite group, let p be a prime and let P ∈ Syl p ( G ) . We prove that P is normal in G if and only if all the irreducible constituents of the permutation character ( 1 P ) G have degree not divisible by p.

Character correspondences in blocks with normal defect groups

Abstract In this paper we give an extension of the Glauberman correspondence to certain characters of blocks with normal defect groups.

A characteristic subgroup and kernels of Brauer characters

If G is finite group and P is a Sylow p-subgroup of G, we prove that there is a unique largest normal subgroup L of G such that L ⋂ P = L ⋂ NG (P). If G is p-solvable, then L is the intersection of the kernels of the irreducible Brauer characters of G of degree not divisible by p.

Primitive characters of subgroups ofM-groups

One of the hardest areas in the Character Theory of Solvable Groups continues to be the monomial groups. A finite group is said to be an M-group (or monomial) if all of its irreducible characters are monomial, that is to say, induced from linear characters. Two are still the main problems on M-groups: are Hall subgroups of M groups monomial? Under certain oddness hypothesis, are normal subgroups of M-groups monomial? In both cases there is evidence that this could be the case: the primitive characters of the subgroups in question are the linear characters. This is the best result up to date ([4], [6]). Recently, some idea appears to be taking form. In [14], T. Okuyama proved that if G is an…

Real class sizes and real character degrees

Perhaps unexpectedly, there is a rich and deep connection between field of values of characters, their degrees and the structure of a finite group. Some of the fundamental results on the degrees of characters of finite groups, as the Ito–Michler and Thompson's theorems, admit a version involving only characters with certain fixed field of values ([DNT, NS, NST2, NT1, NT3]).

ZEROS OF CHARACTERS ON PRIME ORDER ELEMENTS

Suppose that G is a finite group, let χ be a faithful irreducible character of degree a power of p and let P be a Sylow p-subgroup of G. If χ(x) ≠ 0 for all elements of G of order p, then P is cyclic or generalized quaternion. * The research of the first author is supported by a grant of the Basque Government and by the University of the Basque Country UPV 127.310-EB160/98. † The second author is supported by DGICYT.

Nilpotent and abelian Hall subgroups in finite groups

[EN] We give a characterization of the finite groups having nilpotent or abelian Hall pi-subgroups that can easily be verified using the character table.

Squaring a conjugacy class and cosets of normal subgroups

On Brauer’s Height Zero Conjecture

In this paper, the unproven half of Richard Brauer’s Height Zero Conjecture is reduced to a question on simple groups.

Coprime action, characters and decomposition numbers

New Properties of the π-Special Characters

Actions and Invariant Character Degrees

point subgroup. In general, we use the same notation as in 6 and 7 .wx wxPart of the proof of Theorem A depends on the basic properties of theGajendragadkar p-special characters 1 and we assume the reader iswxfamiliar with those. However, we will repeatedly use a deeper fact: anirreducible character a of a Hall p-subgroup

2-Brauer correspondent blocks with one simple module

Abstract One of the main problems in representation theory is to understand the exact relationship between Brauer corresponding blocks of finite groups. The case where the local correspondent has a unique simple module seems key. We study this situation for 2-blocks.

Number of Sylow subgroups in $p$-solvable groups

If G is a finite group and p is a prime number, let vp(G) be the number of Sylow p-subgroups of G. If H is a subgroup of a p-solvable group G, we prove that v p (H) divides v p (G).

CHARACTERS INDUCED FROM FULLY RAMIFIED SUBGROUPS

Suppose that G is a finite π-separable group, let cf(G) be the space of complex class functions of G and let Irr(G) be the set of the irreducible complex characters of G. Let K be an arbitrary Hall...

On the Navarro–Willems conjecture for blocks of finite groups

Abstract We prove that a set of characters of a finite group can only be the set of characters for principal blocks of the group at two different primes when the primes do not divide the group order. This confirms a conjecture of Navarro and Willems in the case of principal blocks.

On the blockwise modular isomorphism problem

As a generalization of the modular isomorphism problem we study the behavior of defect groups under Morita equivalence of blocks of finite groups over algebraically closed fields of positive characteristic. We prove that the Morita equivalence class of a block B of defect at most 3 determines the defect groups of B up to isomorphism. In characteristic 0 we prove similar results for metacyclic defect groups and 2-blocks of defect 4. In the second part of the paper we investigate the situation for p-solvable groups G. Among other results we show that the group algebra of G itself determines if G has abelian Sylow p-subgroups.

Improving the analysis of biogeochemical patterns associated with internal waves in the strait of Gibraltar using remote sensing images

High Amplitude Internal Waves (HAIWs) are physical processes observed in the Strait of Gibraltar (the narrow channel between the Atlantic Ocean and the Mediterranean Sea). These internal waves are generated over the Camarinal Sill (western side of the strait) during the tidal outflow (toward the Atlantic Ocean) when critical hydraulic conditions are established. HAIWs remain over the sill for up to 4 h until the outflow slackens, being then released (mostly) towards the Mediterranean Sea. These have been previously observed using Synthetic Aperture Radar (SAR), which captures variations in surface water roughness. However, in this work we use high resolution optical remote sensing, with the…

Irreducible characters taking root of unity values on $p$-singular elements

In this paper we study finite p-solvable groups having irreducible complex characters chi in Irr(G) which take roots of unity values on the p-singular elements of G.

Weights and Nilpotent Subgroups

In a finite group G, we consider nilpotent weights, and prove a pi-version of the Alperin Weight Conjecture for certain pi-separable groups. This widely generalizes an earlier result by I. M. Isaacs and the first author.

On p-Brauer characters of p′-degree and self-normalizing Sylow p-subgroups

p-Parts of character degrees and the index of the Fitting subgroup

Abstract In a solvable group G, if p 2 does not divide χ ( 1 ) for all χ ∈ Irr ( G ) , then we prove that | G : F ( G ) | p ≤ p 2 . This bound is best possible.

New Refinements of the McKay Conjecture for Arbitrary Finite Groups

Let $G$ be an arbitrary finite group and fix a prime number $p$. The McKay conjecture asserts that $G$ and the normalizer in $G$ of a Sylow $p$-subgroup have equal numbers of irreducible characters with degrees not divisible by $p$. The Alperin-McKay conjecture is a version of this as applied to individual Brauer $p$-blocks of $G$. We offer evidence that perhaps much stronger forms of both of these conjectures are true.

On the Fundamental Theorem of Finite Abelian Groups

(2003). On the Fundamental Theorem of Finite Abelian Groups. The American Mathematical Monthly: Vol. 110, No. 2, pp. 153-154.

Real constituents of permutation characters

Abstract We prove a broad generalization of a theorem of W. Burnside about the existence of real characters of finite groups to permutation characters. If G is a finite group, under the necessary hypothesis of O 2 ′ ( G ) = G , we can also give some control on the parity of multiplicities of the constituents of permutation characters (a result that needs the Classification of Finite Simple Groups). Along the way, we 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 which asks how much information about subgroups of a finite group can be determined by the character table.

A new character correspondence in groups of odd order

Blocks and Normal Subgroups

A McKay bijection for projectors

Inducing characters and nilpotent subgroups

If H H is a subgroup of a finite group G G and γ ∈ Irr ⁡ ( H ) \gamma \in \operatorname {Irr}(H) induces irreducibly up to G G , we prove that, under certain odd hypothesis, F ( G ) F ( H ) \mathbf {F}(G) \mathbf {F}(H) is a nilpotent subgroup of G G .

The Third Main Theorem

Nilpotent and perfect groups with the same set of character degrees

We find a pair of finite groups, one nilpotent and the other perfect, with the same set of character degrees.

Real characters of p′-degree

Regular orbits of hall π-subgroups

Linear characters of Sylow subgroups

Some Open Problems on Coprime Action and Character Correspondences

Brauer's height zero conjecture for the 2-blocks of maximal defect

A reduction theorem for the Galois–McKay conjecture

We introduce H {\mathcal {H}} -triples and a partial order relation on them, generalizing the theory of ordering character triples developed by Navarro and Späth. This generalization takes into account the action of Galois automorphisms on characters and, together with previous results of Ladisch and Turull, allows us to reduce the Galois–McKay conjecture to a question about simple groups.

When is a 𝑝-block a 𝑞-block?

Let p p and q q be distinct prime numbers and let G G be a finite group. If B p B_{p} is a p p -block of G G and B q B_{q} is a q q -block, we study when the set of ordinary irreducible characters in the blocks B p B_{p} and B q B_{q} coincide.

Sylow Normalizers with a Normal Sylow 2-Subgroup

AbstractIf G is a finite solvable group and p is a prime, then the normalizer of a Sylow p-subgroup has a normal Sylow 2-subgroup if and only if all non-trivial irreducible real 2-Brauer characters of G have degree divisible by p.

Self-normalizing Sylow subgroups

Using the classification of finite simple groups we prove the following statement: Let p > 3 p>3 be a prime, Q Q a group of automorphisms of p p -power order of a finite group G G , and P P a Q Q -invariant Sylow p p -subgroup of G G . If C N G ( P ) / P ( Q ) \mathbf {C}_{\mathbf {N}_G(P)/P}(Q) is trivial, then G G is solvable. An equivalent formulation is that if G G has a self-normalizing Sylow p p -subgroup with p > 3 p >3 a prime, then G G is solvable. We also investigate the possibilities when p = 3 p=3 .

p-Brauer characters ofq-defect 0

For ap-solvable groupG the number of irreducible Brauer characters ofG with a given vertexP is equal to the number of irreducible Brauer characters of the normalizer ofP with vertexP. In this paper we prove in addition that for solvable groups one can control the number of those characters whose degrees are divisible by the largest possibleq-power dividing the order of |G|.

A partition of characters associated to nilpotent subgroups

IfG is a finite solvable group andH is a maximal nilpotent subgroup ofG containingF(G), we show that there is a canonical basisP(G|H) of the space of class functions onG vanishing off anyG-conjugate ofH which consists of characters. ViaP(G|H) it is possible to partition the irreducible characters ofG into “blocks”. These behave like Brauerp-blocks and a Fong theory for them can be developed.

A Brauer-Wielandt formula (with an application to character tables)

If a p p -group P P acts coprimely on a finite group G G , we give a Brauer-Wielandt formula to count the number of fixed points | C G ( P ) | | \textbf {C}_{G}(P) | of P P in G G . This serves to determine the number of Sylow p p -subgroups of certain finite groups from their character tables.

Finite Groups with Odd Sylow Normalizers

We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight conjecture for these groups.

p-parts of character degrees

The set of character degrees of a finite group does not determine its solvability

Rational irreducible characters and rational conjugacy classes in finite groups

We prove that a finite group G G has two rational-valued irreducible characters if and only if it has two rational conjugacy classes, and determine the structure of any such group. Along the way we also prove a conjecture of Gow stating that any finite group of even order has a non-trivial rational-valued irreducible character of odd degree.

Bases for induced characters

AbstractIf G is a finite solvable group, we show that Isaacs' theory on partial characters on Hall π-subgroups can be developed for the nilpotent injectors of G. Therefore, the irreducible characters of G are partitioned into blocks associated to some nilpotent subgroups of G.

Characters and Sylow 2-subgroups of maximal class revisited

Abstract We give two ways to distinguish from the character table of a finite group G if a Sylow 2-subgroup of G has maximal class. We also characterize finite groups with Sylow 3-subgroups of order 3 in terms of their principal 3-block.

Brauer correspondent blocks with one simple module

One of the main problems in representation theory is to understand the exact relationship between Brauer corresponding blocks of finite groups. The case where the local correspondent has a unique simple module seems key. We characterize this situation for the principal p-blocks where p is odd.

Defect zero characters predicted by local structure

Let $G$ be a finite group and let $p$ be a prime. Assume that there exists a prime $q$ dividing $|G|$ which does not divide the order of any $p$-local subgroup of $G$. If $G$ is $p$-solvable or $q$ divides $p-1$, then $G$ has a $p$-block of defect zero. The case $q=2$ is a well-known result by Brauer and Fowler.

Degrees of Characters and Values on Prime Order Elements

Two irreducible characters of a finite group with the same value on prime elements have the same degree.

Restriction of odd degree characters and natural correspondences

Let $q$ be an odd prime power, $n > 1$, and let $P$ denote a maximal parabolic subgroup of $GL_n(q)$ with Levi subgroup $GL_{n-1}(q) \times GL_1(q)$. We restrict the odd-degree irreducible characters of $GL_n(q)$ to $P$ to discover a natural correspondence of characters, both for $GL_n(q)$ and $SL_n(q)$. A similar result is established for certain finite groups with self-normalizing Sylow $p$-subgroups. We also construct a canonical bijection between the odd-degree irreducible characters of $S_n$ and those of $M$, where $M$ is any maximal subgroup of $S_n$ of odd index; as well as between the odd-degree irreducible characters of $G = GL_n(q)$ or $GU_n(q)$ with $q$ odd and those of $N_{G}…

The McKay conjecture and Galois automorphisms

The main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group G can be computed locally. The simplest of these conjectures is the ?McKay conjecture? which asserts that the number of irreducible complex characters of G of degree not divisible by p is the same if computed in a p-Sylow normalizer of G. In this paper, we propose a much stronger version of this conjecture which deals with Galois automorphisms. In fact, the same idea can be applied to the celebrated Alperin and Dade conjectures.

VARIATIONS ON THOMPSON'S CHARACTER DEGREE THEOREM

If P is a Sylow- p -subgroup of a finite p -solvable group G , we prove that G^\prime \cap \bf{N}_G(P) \subseteq {P} if and only if p divides the degree of every irreducible non-linear p -Brauer character of G. More generally if π is a set of primes containing p and G is π-separable, we give necessary and sufficient group theoretic conditions for the degree of every irreducible non-linear p -Brauer character to be divisible by some prime in π. This can also be applied to degrees of ordinary characters.

Finite Group Elements where No Irreducible Character Vanishes

AbstractIn this paper, we consider elements x of a finite group G with the property that χ(x)≠0 for all irreducible characters χ of G. If G is solvable and x has odd order, we show that x must lie in the Fitting subgroup F(G).

Induction of Characters and p-Subgroups

Groups with two real Brauer characters

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.

Weights, vertices and a correspondence of characters in groups of odd order

Vertices for characters of $p$-solvable groups

Suppose that G is a finite p-solvable group. We associate to every irreducible complex character X ∈ Irr(G) of G a canonical pair (Q, δ), where Q is a p-subgroup of G and δ ∈ Irr(Q), uniquely determined by X up to G-conjugacy. This pair behaves as a Green vertex and partitions Irr(G) into families of characters. Using the pair (Q, δ), we give a canonical choice of a certain p-radical subgroup R of G and a character η ∈ Irr(R) associated to X which was predicted by some conjecture of G. R. Robinson.

Actions and characters in blocks

HALL SUBGROUPS AND STABLE BRAUER CHARACTERS

AbstractLet $H$ be a Hall $\pi$-subgroup of a finite $\pi$-separable group $G$, and let $\alpha$ be an irreducible Brauer character of $H$. If $\alpha(x)=\alpha(y)$ whenever $x,y \in H$ are $p$-regular and $G$-conjugate, then $\alpha$ extends to a Brauer character of $G$.AMS 2000 Mathematics subject classification: Primary 20C15; 20C20

On irreducible products of characters

Abstract We study the problem when the product of two non-linear Galois conjugate characters of a finite group is irreducible. We also prove new results on irreducible tensor products of cross-characteristic Brauer characters of quasisimple groups of Lie type.

Irreducible restriction and zeros of characters

Let G be a finite group, let N be normal in G and suppose that X is an irreducible complex character of G. Then XN is not irreducible if and only if X vanishes on some coset of N in G.

Characters and generation of Sylow 2-subgroups

Characters and Blocks of Finite Groups

This is a clear, accessible and up-to-date exposition of modular representation theory of finite groups from a character-theoretic viewpoint. After a short review of the necessary background material, the early chapters introduce Brauer characters and blocks and develop their basic properties. The next three chapters study and prove Brauer's first, second and third main theorems in turn. These results are then applied to prove a major application of finite groups, the Glauberman Z*-theorem. Later chapters examine Brauer characters in more detail. The relationship between blocks and normal subgroups is also explored and the modular characters and blocks in p-solvable groups are discussed. Fi…

On Real and Rational Characters in Blocks

Abstract The principal $p$-block of a finite group $G$ contains only one real-valued irreducible ordinary character exactly when $G/{{\bf O}_{p'}(G)}$ has odd order. For $p \ne 3$, the same happens with rational-valued characters. We also prove an analogue for $p$-Brauer characters with $p \geq 3$.

The number of lifts of a Brauer character with a normal vertex

AbstractIn this paper we examine the behavior of lifts of Brauer characters in p-solvable groups. In the main result, we show that if φ∈IBr(G) has a normal vertex Q and either p is odd or Q is abelian, then the number of lifts of φ is at most |Q:Q′|. As a corollary, we prove that if φ∈IBr(G) has an abelian vertex subgroup Q, then the number of lifts of φ in Irr(G) is at most |Q|.

Characters of 𝑝’-degree with cyclotomic field of values

If p p is a prime number and G G is a finite group, we show that G G has an irreducible complex character of degree not divisible by p p with values in the cyclotomic field Q p \mathbb {Q}_p .

A characterisation of nilpotent blocks

Let $B$ be a $p$-block of a finite group, and set $m=$ $\sum \chi(1)^2$, the sum taken over all height zero characters of $B$. Motivated by a result of M. Isaacs characterising $p$-nilpotent finite groups in terms of character degrees, we show that $B$ is nilpotent if and only if the exact power of $p$ dividing $m$ is equal to the $p$-part of $|G:P|^2|P:R|$, where $P$ is a defect group of $B$ and where $R$ is the focal subgroup of $P$ with respect to a fusion system $\CF$ of $B$ on $P$. The proof involves the hyperfocal subalgebra $D$ of a source algebra of $B$. We conjecture that all ordinary irreducible characters of $D$ have degree prime to $p$ if and only if the $\CF$-hyperfocal subgrou…

Finite Groups with Only One NonLinear Irreducible Representation

Let 𝕂 be an algebraically closed field. We classify the finite groups having exactly one irreducible 𝕂-representation of degree bigger than one. The case where the characteristic of 𝕂 is zero, was done by G. Seitz in 1968.

Restriction of characters to Sylow normalizers

Suppose that G is a finite p -solvable group and let \chi \in {\rm Irr}(G) be of p^\prime -degree. In this note, we investigate when \chi remains irreducible when restricted to {\bf {N)}_{G}(P) .

Degrees of rational characters of finite groups

Abstract A classical theorem of John Thompson on character degrees states that if the degree of any complex irreducible character of a finite group G is 1 or divisible by a prime p, then G has a normal p-complement. In this paper, we consider fields of values of characters and prove some improvements of this result.

Coprime Actions, Fixed-Point Subgroups and Irreducible Induced Characters

Characters with stable irreducible constituents

Decomposition numbers and local properties

Abstract If G is a finite group and p is a prime, we give evidence that the p-decomposition matrix encodes properties of p-Sylow normalizers.

On fully ramified Brauer characters

Let Z be a normal subgroup of a finite group, let p≠5 be a prime and let λ∈IBr(Z) be an irreducible G-invariant p-Brauer character of Z. Suppose that λG=eφ for some φ∈IBr(G). Then G/Z is solvable. In other words, a twisted group algebra over an algebraically closed field of characteristic not 5 with a unique class of simple modules comes from a solvable group.

Sylow subgroups, exponents, and character values

If G G is a finite group, p p is a prime, and P P is a Sylow p p -subgroup of G G , we study how the exponent of the abelian group P / P ′ P/P’ is affected and how it affects the values of the complex characters of G G . This is related to Brauer’s Problem 12 12 . Exactly how this is done is one of the last unsolved consequences of the McKay–Galois conjecture.

A submatrix of the character table

Let G be a finite group and let p be a prime number. We consider the Submatrix of the character table of G whose rows are indexed by the characters in blocks of maximal defect, and whose columns are indexed by the conjugacy classes of P′-size. We prove that this matrix has maximum rank.

HEIGHTS OF CHARACTERS IN BLOCKS OF $p$-SOLVABLE GROUPS

In this paper, it is proved that if $B$ is a Brauer $p$ -block of a $p$ -solvable group, for some odd prime $p$ , then the height of any ordinary character in $B$ is at most $2b$ , where $p^b$ is the largest degree of the irreducible characters of the defect group of $B$ . Some other results that relate the heights of characters with properties of the defect group are obtained.

Fields of values of odd-degree irreducible characters

Abstract In this paper we clarify the quadratic irrationalities that can be admitted by an odd-degree complex irreducible character χ of an arbitrary finite group. Write Q ( χ ) to denote the field generated over the rational numbers by the values of χ, and let d > 1 be a square-free integer. We prove that if Q ( χ ) = Q ( d ) then d ≡ 1 (mod 4) and if Q ( χ ) = Q ( − d ) , then d ≡ 3 (mod 4). This follows from the main result of this paper: either i ∈ Q ( χ ) or Q ( χ ) ⊆ Q ( exp ⁡ ( 2 π i / m ) ) for some odd integer m ≥ 1 .

Invariant characters and coprime actions on finite nilpotent groups

Suppose that a group A acts via automorphisms on a nilpotent group G having coprime order. Given an A-invariant character \(\chi \in {\rm Irr}(G)\), we show that the A-primitive irreducible characters that induce \(\chi \) from an A-invariant subgroup of G all have equal degree. We use this result to obtain some information about the characters of groups of p-length 1.

Order of products of elements in finite groups

If G is a finite group, p is a prime, and x∈G, it is an interesting problem to place x in a convenient small (normal) subgroup of G, assuming some knowledge of the order of the products xy, for certain p‐elements y of G.

Groups whose real irreducible characters have degrees coprime to p

Abstract In this paper we study groups for which every real irreducible character has degree not divisible by some given odd prime p .

Sylow Normalizers and Brauer Character Degrees

Suppose that G is a finite group. In this note, we show that a local condition about Sylow normalizers is equivalent to a global condition on the degrees of certain irreducible Brauer characters of G. Theorem A. Let G be a finite ”p; q•-solvable group, and let Q ∈ SylqG‘ and P ∈ SylpG‘. Then every irreducible p-Brauer character of G of q′degree has p′-degree if and only if NGQ‘ is contained in some G-conjugate of NGP‘. Theorem A needs a solvability hypothesis. If p = 7, then the irreducible p-Brauer characters of the group G = PSL2; 27‘ have degrees ”1; 13; 26; 28•. If we set q = 2, then each q′-degree is also a p′-degree.

Abelian Sylow subgroups in a finite group, II

Abstract Let p ≠ 3 , 5 be a prime. We prove that Sylow p-subgroups of a finite group G are abelian if and only if the class sizes of the p-elements of G are all coprime to p. This gives a solution to a problem posed by R. Brauer in 1956 (for p ≠ 3 , 5 ).

Coprime actions and correspondences of Brauer characters

We prove several results giving substantial evidence in support of the conjectural existence of a Glauberman–Isaacs bijection for Brauer characters under a coprime action. We also discuss related bijections for the McKay conjecture.

On a question of C. Bonnafé on characters and multiplicity free constituents

Abstract In 2006, C. Bonnafe posed a general question on characters of finite groups. A positive answer would have reduced drastically some proofs by G. Lusztig.

Quadratic characters in groups of odd order

Abstract We prove that in a finite group of odd order, the number of irreducible quadratic characters is the number of quadratic conjugacy classes.

Partial characters with respect to a normal subgroup

AbstractSuppose that G is a π-separable group. Let N be a normal π1-subgroup of G and let H be a Hall π-subgroup of G. In this paper, we prove that there is a canonical basis of the complex space of the class functions of G which vanish of G-conjugates ofHN. When N = 1 and π is the complement of a prime p, these bases are the projective indecomposable characters and set of irreduciblt Brauer charcters of G.

Groups with exactly one irreducible character of degree divisible byp

Let [math] be a prime. We characterize those finite groups which have precisely one irreducible character of degree divisible by [math] .

Blocks with 𝑝-power character degrees

Let B B be a p p -block of a finite group G G . If χ ( 1 ) \chi (1) is a p p -power for all χ ∈ Irr ⁡ ( B ) \chi \in \operatorname {Irr}(B) , then B B is nilpotent.

Rationality and normal 2-complements

Abstract We study the finite groups in which every irreducible rational valued character is linear, and those in which every rational element is central.

p-Parts of Brauer character degrees

Abstract Let G be a finite group and let p be an odd prime. Under certain conditions on the p-parts of the degrees of its irreducible p-Brauer characters, we prove the solvability of G. As a consequence, we answer a question proposed by B. Huppert in 1991: If G has exactly two distinct irreducible p-Brauer character degrees, then is G solvable? We also determine the structure of non-solvable groups with exactly two irreducible 2-Brauer character degrees.

Conjugacy class numbers and π-subgroups

Character degrees, derived length and Sylow normalizers

Let P be a Sylow p-subgroup of a monomial group G. We prove that dl $ ({\Bbb N}_G (P)/P') $ is bounded by the number of irreducible character degrees of G which are not divisible by p.