In this paper, we prove one direction of a conjecture of Navarro-Rizo-Schaeffer Fry-Vallejo positing an algorithm to determine from the character table whether a finite group has $2$-generated Sylow $3$-subgroups. This gives further evidence of the blockwise version of the Galois-McKay conjecture (also known as the Alperin-McKay-Navarro conjecture). A key step involves proving the Isaacs-Navarro Galois conjecture for principal blocks for finite groups with a certain structure.
We characterize finite groups having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block for p=2,3. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay-Navarro conjecture.
G. Navarro has conjectured a necessary and sufficient condition for a finite group $G$ to have a self-normalizing Sylow 2-subgroup, which is given in terms of the ordinary irreducible characters of $G$. In a previous article, the author has reduced the proof of this conjecture to showing that certain related statements hold for simple groups. In this article, we describe the action of Galois automorphisms on the Howlett-Lehrer parametrization of Harish-Chandra induced characters. We use this to complete the proof of the conjecture by showing that the remaining simple groups satisfy the require…
We describe the Sylow subgroups of Gal(Q) for an odd prime p, by observing and studying their decomposition as a semidirect product of Z_p acting on F, where F is a free pro-p group, and Z_p are the p-adic integers. We determine the finite Z_p-quotients of F and more generally show that every split embedding problem of Z_p-groups for F is solvable. Moreover, we analyze the Z_p-action on generators of F.
We prove that splendid Morita equivalences between principal blocks of finite groups with dihedral Sylow $2$-subgroups realised by Scott modules can be lifted to splendid Morita equivalences between principal blocks of finite groups with generalised quaternion Sylow $2$-subgroups realised by Scott modules.
$\DeclareMathOperator{\Aut}{Aut}$Let $p, q$ be distinct primes, with $p > 2$. We classify the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, such that the Sylow $p$-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are cyclic. This is equivalent to classifying the skew braces $(G, \cdot, \circ)$. Furthermore, we prove that if $G$ and $Γ$ are groups of order $p^{2} q$ with non-is…
We have found a list of finite simple groups with cyclic Sylow $p$-subgroup whose principal $p$-blocks have Brauer trees in the shape of a star, that is a tree of diameter at most $2$. Moreover, for an arbitrary finite group $G$ with cyclic Sylow $p$-subgroup, we have obtained a necessary condition when the Brauer tree of the principal $p$-block of $G$ is a star.
Let $G$ be a finite group with a Sylow $p$-subgroup $P$. We prove that the principal $p$-blocks of $G$ and $N_G(P)$ are perfectly isometric under the assumption $G$ has a cyclic $p$-hyperfocal subgroup.
In this note, we prove that if every character of a finite group $G$ fixed by an order-2 Galois automorphism has odd degree, then $G$ has a normal Sylow $2$-subgroup. On the way, we study extensions of characters of $GL_n(q)$, $q$ odd, to the group extended by the transpose-inverse automorphism and prove that unipotent characters of $PSL_n(q)$ extend to rational characters of its automorphism group.
We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.
For $Γ$ a group of order $mp$ for $p$ prime where $gcd(p,m)=1$, we consider those regular subgroups $N\leq Perm(Γ)$ normalized by $λ(Γ)$, the left regular representation of $Γ$. These subgroups are in one-to-one correspondence with the Hopf-Galois structures on separable field extensions $L/K$ with $Γ=Gal(L/K)$. This is a follow up to the author's earlier work where, by assuming $p>m$, one has that all such $N$ lie within the normalizer of the $p$-Sylow subgroup of $λ(Γ)$. Here we show that one only need assume that all groups of a given order $mp$ have a unique $p$-Sylow subgroup, and that $p…
For a prime $p$, we determine a Sylow $p$-subgroup $D$ of a finite group $G$ such that the principal $p$-block $B$ of $G$ has four irreducible ordinary characters. It has been determined already for the cases where the number is up to three by work by R. Brauer, J. Brandt, and V.A. Belonogov thirty years ago. Our proof relies on the classification of finite simple groups.
For a prime $\ell$, the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with $\ell'$-degree and the corresponding set for the normalizer of a Sylow $\ell$- subgroup. Navarro's refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro--Spaeth--Vallejo have reduced the McKay--Navarro conjecture to certain "inductive" conditions on finite simple groups. We prove that these inductive McKay--Navarro (also c…
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.
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 …
We give a new approach to the construction of derived equivalences between blocks of finite groups, based on perverse equivalences, in the setting of Broué's conjecture. We provide in particular local and global perversity data describing the principal blocks and the derived equivalences for a number of finite simple groups with Sylow subgroups C3xC3. We also examine extensions to automorphism groups in a general setting.
Let $G:={^2G_2}(q)$ be the simple Ree group with $q=3^{2k+1}$ and $k$ a positive integer. We show that the centre of the principal block $Z(kGe_0)$, where $k$ is an algebraically closed field of characteristic $3$, is not isomorphic to the centre of the Brauer corresponding block $Z(kN_G(P))$, where $N_G(P)$ is the normaliser in $G$ of a Sylow $3$-subgroup. As part of the proof, we compute the conjugacy classes of elements and the character tables of the maximal parabolic subgroups of $G$.
The structure of the Galois group of the maximal unramified p-extension of an imaginary quadratic field is restricted in various ways. In this paper we construct a family of finite 3-groups satisfying these restrictions. We prove several results about this family and characterize them as finite extensions of certain quotients of a Sylow pro-3 subgroup of SL_2(Z_3). We verify that the first group in the family does indeed arise as such a Galois group and provide a small amount of evidence that this may hold for the other members. If this were the case then it would imply that there is no upper …
Eaton and Moretó proposed an extension of Brauer's famous height zero conjecture on blocks of finite groups to the case of non-abelian defect groups, which predicts the smallest non-zero height in such blocks in terms of local data. We show that their conjecture holds for principal blocks of quasi-simple groups, for all blocks of finite reductive groups in their defining characteristic, as well as for all covering groups of symmetric and alternating groups. For the proof, we determine the minimal non-trivial character degrees of Sylow $p$-subgroups of finite reductive groups in characteristic~…
Let $S_5$ denote the symmetric group on 5 letters, and let $\hat{S}_5$ denote a non-trivial double cover of $S_5$ whose Sylow 2-subgroups are generalized quaternion. We determine the universal deformation rings $R(S_5,V)$ and $R(\hat{S}_5,V)$ for each mod 2 representation $V$ of $S_5$ that belongs to the principal 2-modular block of $S_5$ and whose stable endomorphism ring is given by scalars when it is inflated to $\hat{S}_5$. We show that for these $V$, a question raised by the first author and Chinburg concerning the relation of the universal deformation ring of $V$ to the Sylow 2-subgroups…
We show that for every finitely generated closed subgroup $K$ of a non-solvable Demushkin group $G$, there exists an open subgroup $U$ of $G$ containing $K$, and a continuous homomorphism $τ\colon U \to K$ satisfying $τ(k) = k$ for every $k \in K$. We prove that the intersection of a pair of finitely generated closed subgroups of a Demushkin group is finitely generated (giving an explicit bound on the number of generators). Furthermore, we show that these properties of Demushkin groups are preserved under free pro-$p$ products, and deduce that Howson's theorem holds for the Sylow subgroups of …
A Bloch-Kato pro-p group G is a pro-p group with the property that the F_p-cohomology ring of every closed subgroup of G is quadratic. It is shown that either such a pro-p group G contains no closed free pro-p groups of infinite rank, or there exists an orientation $θ\colon G\rightarrow \Z_p^\times$ such that G is theta-abelian. In case that G is also finitely generated, this implies that G is powerful, p-adic analytic with d(G)=cd(G), and its \F_p-cohomology ring is an exterior algebra. These results will be obtained by studying locally powerful groups (see Theorem A). There are certain Galoi…
In [7], G. Navarro proposed a refinement of the McKay conjecture involving a special class of Galois automorphisms. In [6] this new conjecture was verified by the author for the alternating groups A(n) when p=2. In this note the Navarro conjecture is verified for the p-singular characters of the principal block of the alternating groups in the case of abelian defect when p is odd.
We initiate the study of profinite groups of non-negative deficiency. The principal focus of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group $G$ of non-negative deficiency gives rather strong consequences for the structure of $G$. To make this precise we introduce the notion of $p$-deficiency ($p$ a prime) for a profinite group $G$. This concept is more useful in the study of profinite groups then the notion of deficiency. We prove that if the $p$-deficiency of $G$ is positive and $N$ is a finitely generated normal subgroup…
The classical Galois theory deals with certain finite algebraic extensions and establishes a bijective order reversing correspondence between the intermediate fields and the subgroups of a group of permutations called the Galois group of the extension. It has been the dream of many mathematicians at the end of the nineteenth century to generalize these results to systems of algebraic partial differential (PD) equations and the corresponding finitely generated differential extensions, in order to be able to add the word differential in front of any classical statement. The achievement of the Pi…
