In 1998, the second author of this paper raised the problem of classifying the irreducible characters of Sn of prime power degree. Zalesskii proposed the analogous problem for quasi-simple groups, and he has, in joint work with Malle, made substantial progress on this latter problem. With the exception of the alternating groups and their double covers, their work provides a complete solution. In this article we first classify all the irreducible characters of Sn of prime power degree (Theorem 2.4), and then we deduce the corresponding classification for the alternating groups (Theorem 5.1), thus providing the answer for one of the two remaining families in Zalesskii's problem. This classification has another application in group theory. With it, we are able to answer, for alternating groups, a question of Huppert: which simple groups G have the property that there is a prime p for which G has an irreducible character of p-power degree > 1 and all of the irreducible characters of G have degrees that are relatively prime to p or are powers of p?

The case of the double covers of the symmetric and alternating groups will be dealt with in a forthcoming paper; in particular, this completes the answer to Zalesskii's problem.

The paper is organized as follows. In Section 2, some results on hook lengths in partitions are proved. These results lead to an algorithm which allows us to show that every irreducible representation of Sn with prime power degree is labelled by a partition having a large hook. In Section 3, we obtain a new result concerning the prime factors of consecutive integers (Theorem 3.4). In Section 4 we prove Theorem 2.4, the main result. To do so, we combine the algorithm above with Theorem 3.4 and work of Rasala on minimal degrees. This implies Theorem 2.4 for large n. To complete the proof, we check that the algorithm terminates appropriately for small n (that is, those n [les ] 9.25 · 108) with the aid of a computer. In the last section we derive the classification of irreducible characters of An of prime power degree, and we solve Huppert's question for alternating groups.

Results from the character theory of symmetric groups are used to obtain an asymptotic estimate for the subgroup growth of fundamental groups of closed 2-manifolds. The main result implies an affirmative answer, for the class of groups investigated, to a question of Lubotzky's concerning the relationship between the subgroup growth of a one-relator group and that of a free group of appropriately chosen rank. As byproducts, an interesting statistical property of commutators in symmetric groups and the fact that in a ‘large’ surface group almost all finite index subgroups are maximal are obtained, among other things. The approach requires an asymptotic estimate for the sum $\Sigma 1/(\chi_\lambda(1))^s$ taken over all partitions $\lambda$ of $n$ with fixed $s \ge 1$ , which is also established.

Let F be a meromorphic function in the complex plane. We investigate the behaviour of the iterates of F in a Baker domain B. In particular, we describe the dynamics of the orbits with the help of conformal conjugacies; that is, we determine a function φ which is univalent in a large simply connected subdomain of B such that φ(F(z))=T(φ(z)) holds throughout B. Here T is either a parabolic or hyperbolic Möbius transformation mapping either a half plane or [Copf ] onto itself. This functional equation is always solvable in a Baker domain if F has only finitely many poles. Moreover, there is an example of a function with infinitely many poles where one cannot find an appropriate conformal conjugacy in an invariant Baker domain.

The closed 3-manifolds of constant positive curvature were classified long ago by Seifert and Threlfall. Using well-known information about the orthogonal group O(4), their full isometry groups Isom(M) are calculated. It is determined which elliptic 3-manifolds admit Seifert fiberings that are invariant under all isometries, and it is verified that the inclusion of Isom(M) to Diff(M) is a bijection on path components.

We extend uncertainty principles which are valid for the Fourier transform to the setting of the ambiguity function. A general result is established for annihilating sets: strongly/weakly annihilating sets for the Fourier transform yield such sets for the ambiguity function, extending a result known for sets of finite measure. We apply this to sublevel sets of nondegenerate quadratic forms. Our main result is a sharp version of Beurling's uncertainty principle for the ambiguity function.

An independence algebra is an algebra A in which the subalgebras satisfy the exchange axiom, and any map from a basis of A into A extends to an endomorphism. Independence algebras fall into two classes; the first is specified by a set X, a group G, and a G-space C. The second is much more restricted; it is shown that the subalgebra lattice is a projective or affine geometry, and a complete classification of the finite algebras is given.

We are interested in Weil representations of Sp(2n, R), where R is the ring Z/plZ, p is an odd prime and l is a positive integer, or, more generally, R = [Oscr ]/[pfr ]l, where [Oscr ] is the ring of integers of a local field, [pfr ] is the maximal ideal of [Oscr ] and [Oscr ]/[pfr ] has odd characteristic. One reason for this interest is that a continuous finite-dimensional complex representation of Sp(2n, [Oscr ]) has to factor through a representation of Sp(2n, [Oscr ]/[pfr ]l) for some l.

Every convolution operator on a space of ultradifferentiable functions of Beurling or Roumieu type and on the corresponding space of ultradistributions is hypercyclic and chaotic when it is not a multiple of the identity. The operator of differentiation is hypercyclic on the space A−∞, but it need not be hypercyclic on radial weighted algebras of entire functions.

We consider the classical universal covering $\exp: {{\mathbb C}}\To {{\mathbb C}}^*$ of the complex torus as an algebraic structure. The exponentiation is seen here as an abstract homomorphism from a divisible torsion-free group onto the multiplicative group of an algebraically closed field of characteristic zero, with cyclic kernel. We prove that any structure satisfying this description is isomorphic to the classical one provided that the cardinality of the underlying field is equal to that of ${{\mathbb C}}$. This can also be seen as a model-theoretic statement on the categoricity of a corresponding $L_{\omega_1,\omega}$-sentence. The proof is a combination of arithmetic and model-theoretic methods.

S.-Y. Cheng, P. Li and S.-T. Yau proved comparison theorems for the volume of extrinsic balls in minimal submanifolds of space forms. These results were extended by S. Markvorsen for minimal submanifolds of a riemannian manifold with just an upper bound on the sectional curvature. In the paper an isoperimetric inequality for extrinsic balls in minimal submanifolds of a riemannian manifold N with sectional curvatures bounded from above by a non-positive constant is found. As a corollary of this result an alternative proof is obtained of the comparison for the volume of extrinsic balls stated by the preceding authors, but now the equality case is characterized when the upper bound for the sectional curvatures of the ambient manifold is strictly negative. Finally, when the sectional curvatures of N are bounded from above for any constant (positive or negative), it is proved that the ∞-isoperimetric quotient of the extrinsic balls is bounded from below by the mean curvature of the geodesic spheres in the m-dimensional real space forms.

We prove a general Borg-type result for reflectionless unitary CMV operators $U$ associated with orthogonal polynomials on the unit circle. The spectrum of $U$ is assumed to be a connected arc on the unit circle. This extends a recent result of Simon in connection with a periodic CMV operator with spectrum the whole unit circle.

In the course of deriving the Borg-type result we also use exponential Herglotz representations of Caratheodory functions to prove an infinite sequence of trace formulas connected with the CMV operator $U$.

In this article we extend well-known results of Livsic on the regularity of measurable solutions to the cocycle equation. Livsic proved a regularity result for real valued cocycles, but for compact Lie group valued cocycles his result was restricted to coboundaries (that is, trivial or constant cocycles). In this article we prove the complete result.

We present several useful applications of these results.

It is shown that if S is an aperiodic random walk on the integers, S* is the Markov chain that arises when S is killed when it leaves the non-negative integers, and H+ is the renewal process of weak increasing ladder heights in S, then there is a 1[ratio ]1 correspondence between functions which are non-negative and superregular for S* and H+. This allows all the regular functions for S* to be described, and thus a result due to Spitzer to be completed for the recurrent case. This result is then applied to give a ratio limit theorem for Px(τ* =n)/P0{τ*=n}, where τ* is the lifetime of S*, in the case when S drifts to −∞, and the right-hand tail of its step distribution is ‘locally sub-exponential’.

It is proved that the measure algebra M(G) of a locally compact group G is Connes-amenable if and only if G is amenable.

We work in set theory without the axiom of choice: ZF. We show that the axiom BC: Compact Hausdorff spaces are Baire, is equivalent to the following axiom: Every tree has a subtree whose levels are finite, which was introduced by Blass (cf. [4]). This settles a question raised by Brunner (cf. [9, p. 438]). We also show that the axiom of Dependent Choices is equivalent to the axiom: In a Hausdorff locally convex topological vector space, convex-compact convex sets are Baire. Here convex-compact is the notion which was introduced by Luxemburg (cf. [16]).

It is proved that all correlations of the sequence of Farey fractions exist and explicit formulas are provided for the correlation measures.

Further properties of a group $\Gamma$ introduced by the first author in 1980 (sometimes called the first Grigorchuk group) are established. These are notably that any two infinite finitely generated subgroups of $\Gamma$ have isomorphic subgroups of finite index and that the generalized word problem for $\Gamma$ is soluble.

The paper deals with the estimates of the blow-up rate and the blow-up set of positive solutions to semilinear parabolic systems coupled in the equations and boundary conditions. The explicit upper and lower bounds of the blow-up rate are obtained. Moreover, it is proved that the blow-up set contains a single point x = 1 under the proper conditions on the initial data.

The well-known Oberwolfach problem is to show that it is possible to 2-factorize Kn (n odd) or Kn less a 1-factor (n even) into predetermined 2-factors, all isomorphic to each other; a few exceptional cases where it is not possible are known. A completely new technique is introduced that enables it to be shown that there is a solution when each 2-factor consists of k r-cycles and one (n−kr)-cycle for all n [ges ] 6kr−1. Solutions are also given (with three exceptions) for all possible values of n when there is one r-cycle, 3 [les ] r [les ] 9, and one (n−r)-cycle, or when there are two r-cycles, 3 [les ] r [les ] 4, and one (n−2r)-cycle.

Various notions of dimension for discrete groups are compared. A group is exhibited that acts with finite stabilizers on an acyclic 2-complex in such a way that the fixed point subcomplex for any non-trivial finite subgroup is contractible, but such that the group does not admit any such action on a contractible 2-complex. This group affords a counterexample to a natural generalization of the Eilenberg–Ganea conjecture.