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Let Gbe a finite group, let Hbe a core-free subgroup and let b(G, H) denote the base size for the action of Gon G/H. Let α(G)be the number of conjugacy classes of core-free subgroups Hof Gwith b(G,H)≥3. We say that Gis a strongly base-two group if α(G)≤1, which means that almost every faithful transitive permutation representation of Ghas base size 2. In this paper, we study the strongly base-two finite groups with trivial Frattini subgroup.

If His a Hall subgroup of a finite group G, it was proven in 1989 using the classification of finite simple groups that all the irreducible complex characters of Hextend to Gif and only if there is N⊴Gsuch that HN=Gand H∩N=1. In this note we offer a modular version of this result under more general hypotheses.

We prove that if Gis a finite simple group and x,y∈Gare involutions, then |xG∩CG(y)|→∞as |G|→∞. This extends results of Guralnick–Robinson and Skresanov. We also prove a related result about CG(t)/O(CG(t))that does not require the classification of finite simple groups.

We prove some results about closures of certain matrix varieties consisting of elements with the same centralizer dimension. This generalizes a result of Dixmier and has applications to topological generation of simple algebraic groups.

We give a sharp divisibility bound, in terms of g, for the degree of the field extension required to realize the endomorphisms of an abelian variety of dimension gover an arbitrary number field; this refines a result of Silverberg. This follows from a stronger result giving the same bound for the order of the component group of the Sato–Tate group of the abelian variety, which had been proved for abelian surfaces by Fité–Kedlaya–Rotger–Sutherland. The proof uses Minkowski’s reduction method, but with some care required in the extremal cases when pequals 2 or a Fermat prime.

Advances in biodiversity science, coupled with new technologies and big data platforms, are expanding our ability to explore and understand the natural world. For the first time, biologists can link data from growing repositories and computational approaches to better integrate plant evolution and ecology at the broadest extents. The emerging synthesis is reshaping our views of plant diversification and guiding new approaches to conservation.

Animal colour is a complex trait shaped by multiple selection pressures that can vary across geography. The thermal melanism hypothesis predicts that darker coloration is beneficial to animals in colder regions because it allows for more rapid solar absorption. Here, we use community science images of three closely related species of North American ratsnakes (genus Pantherophis) to examine if climate predicts colour variation across range-wide scales. We predicted that darker individuals are found in colder regions and higher elevations, in accordance with the thermal melanism hypothesis. Using an unprecedented dataset of over 8000 images, we found strong support for temperature as a key predictor of darker colour, supporting thermal melanism. We also found that elevation and precipitation are predictive of colour, but the direction and magnitude of these effects were more variable across species. Our study is the first to quantify colour variation in Pantherophisratsnakes, highlighting the value of community science images for studying range-wide colour variation.

The notion of adequate subgroups was introduced by Thorne (J Inst Math Jussieu. arXiv:1107.5989, to appear). It is a weakening of the notion of big subgroup used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to prove some new lifting theorems. It was shown in Guralnick et al. (J Inst Math Jussieu. arXiv:1107.5993, to appear) that certain groups were adequate. One of the key aspects was the question of whether the span of the semisimple elements in the group is the full endomorphism ring of an absolutely irreducible module. We show that this is the case in prime characteristic pfor p-solvable groups as long the dimension is not divisible by p. We also observe that the condition holds for certain infinite groups. Finally, we present the first examples showing that this condition need not hold and give a negative answer to a question of Richard Taylor.

AbstractWe study Frobenius groups acting on curves.

For each finite simple group Gthere is a conjugacy class CGsuch that each nontrivial element of Ggenerates Gtogether with any of more than 1/10 of the members of CG. Precise asymptotic results are obtained for the probability implicit in this assertion. Similar results are obtained for almost simple groups.

Let Abe a transitive subgroup of Sn. We show that the largest cyclic quotient of Ahas order at most n. This can be interpreted as an equivalent result about extensions of constants in the Galois closure of a covering of curves over a finite field. We also prove that the point stabilizer in a finite primitive permutation group always has a faithful orbit.

In this note, we show that the set of all commuting d -tuples of commuting n×n matrices that are contained in an n -dimensional commutative algebra is a closed set, and therefore, Gerstenhaber's theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also study the variety of n -dimensional commutative subalgebras of M n (F) , and show that it is irreducible of dimension n−n for n4 , but reducible, of dimension greater than n−n for n7 .

LetKbe a finite field of characteristicp. A polynomialfwith coefficients inKis said to beexceptionalif it induces a permutation on infinitely many finite extensions ofK. Lettbe a transcendental, andK̄be an algebraic closure ofK. The exceptional polynomials known to date are constructed from certain twists of polynomialsg, such that the splitting field ofg(X)−toverK̄is rational. On the other hand we know that except forp=2 or 3, either the degree of an exceptional polynomialfis a power of the characteristicp, orfis a Dickson polynomial. The case degf=phas been settled completely. We give a new series of exceptional polynomials of degreepmwithmeven which does not follow the above mentioned construction principle. We show under certain additional hypotheses that the associated monodromy groups of exceptional polynomials will be severely restricted. In particular, we determine precisely what the geometric monodromy groups of exceptional polynomials of degreep2will be.

AbstractLet Kbe an algebraically closed field of characteristic zero, and Sa nonempty subset of Ksuch that S Q= ? and card S< card K, where Qis the field of rational numbers. By Zorn's Lemma, there exist subfields Fof Kwhich are maximal with respect to the property of being disjoint from S. This paper examines such subfields and investigates the Galois group Gal K/F along with the lattice of intermediate subfields.

Weakenings of the L and P-properties are considered. It ib shown that if semigroups, rings, or Lie Algebras of matrices have certain of these properties, then they are triangular. In particular the result of Zassenhaus on L-semigroups is generalized. Also, Lie Algebras over a field of characteristic 0 which have the L-property are triangular.

Let V = [image omitted] . We consider two groups of unitary operators on V, namely[image omitted]  acting on V by conjugation, and [image omitted]  acting on V via U:A→UAU1. We study the overgroups of these unitary groups in GL(V). The results are then applied to classify the invertible linear maps on V and various subspaces of V which preserve some unitary similarity invariant or unitary congruence invariant property. This provides a unified proof for various existing results, and answers a number of open questions on linear preserver problems.

Let F be an algebraically closed field. Identify PSLn(F) as a subgroup of SLn2(F) via the conjugation representation on Mn(F). We find all overgroups of PSLn(F). This was studied by Dokovic and Platonov in characteristic zero. The results are then applied to classify the invertible linear maps on Mn(F)or sln(F) which preserve some similarity invariant property. This provides a unified proof for various existing results and yields some new results. We consider similar questions for actions on symmetric and exterior spaces.

We improve a result of Liebeck and Saxl concerning the minimal degree of a primitive permutation group and use it to strengthen a result of Guralnick and Neubauer on generic covers of Riemann surfaces.

It is shown that if $ G$, then any abelian quotient of $ G$. This was motivated by a question in Galois theory. The field theoretic interpretation of the result is that if $ M/K$ is an abelian extension contained in the normal closure of $ M$ is at most the degree of .

We give a short proof of the Motzkin-Taussky result that the variety of commuting pairs of matrices is irreducible. An easy consequence of this is that any two generated commutative subalgebra of n×n matrices has dimension at most n. We also answer an old question of Gerstenhaber by showing that the set of commuting triples of n×n matrices is not irreducible for n≥32.

Let A ε Mn(D) with D a division ring. It is shown that unless A satisfies a central irreducible polynomial there exist XkεMn(D) so that rank (XkA - AXk) = k for any k ≤ 2 min {rank(A - cI)∣c ε Z(D)}. The possible ranks of commutators are determined in the other case.

We prove that, over a PID, if two matrices $ {\mathbf{A}}$ $ {\mathbf{B}}$, then $ {\mathbf{A}}$ $ {\mathbf{B}}$-theory of noetherian rings.

We show that direct-sum cancellation can fail for modules in a single genus, over commutative noetherian rings of dimension 1. The example provides an application of noncommutative methods (including "wild representation type") to commutative ring theory.

Let $ K/k$ is not an algebraic extension of a finite field. Let $ {K^{\ast} }$, and let $ \Theta (K/k)$ $ \Gamma = {K^{\ast} }/\Theta (K/k)$. For algebraic number fields and function fields, we give a complete classification of those $ G$ is nontrivial.

In this article, we attempt to generalize the result that for a commutative ring $ R$-automorphisms of $ {M_n}(R)$. It is shown that a slightly weaker stable version of the result is still valid for affine semiprime noetherian pi rings. We also show that the automorphism group of an affine commutative domain of positive dimension acts faithfully on the spectrum of the domain. We investigate other questions involving bimodules and automorphisms and extend a result of Smith on the first Weyl algebra as a fixed ring.

In this paper we obtain the structure of locally finite groups which have periodic cohomology after some steps.

Worldwide decline in biodiversity during the Holocene has impeded a comprehensive understanding of pre-human biodiversity and biogeography. This is especially true on islands, because many recently extinct island taxa were morphologically unique, complicating assessment of their evolutionary relationships using morphology alone. The Caribbean remains an avian hotspot but was more diverse before human arrival in the Holocene. Among the recently extinct lineages is the enigmatic genus Nesotrochis,comprising three flightless species. Based on morphology, Nesotrochishas been considered an aberrant rail (Rallidae) or related to flufftails (Sarothruridae). We recovered a nearly complete mitochondrial genome of Nesotrochis steganinosfrom fossils, discovering that it is not a rallid but instead is sister to Sarothruridae, volant birds now restricted to Africa and New Guinea, and the recently extinct, flightless Aptornithidae of New Zealand. This result suggests a widespread or highly dispersive most recent common ancestor of the group. Prior to human settlement, the Caribbean avifauna had a far more cosmopolitan origin than is evident from extant species.

We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al.‘Character levels and character bounds for finite classical groups’, Preprint, 2019, arXiv:1904.08070] by different methods.