A structure theorem for right invariant right holoids

Communicated by Boris M. Schein     

Solving equations in groups: A survey of Frobenius' theorem

A fundamental result of Frobenius states that in a finite group the number of elements which satisfy the equationx ⁿ=1, wheren divides the order of the group, is divisible byn. Here 1 denotes the identity of the group. This theorem and several generalizations were obtained by Frobenius at the turn of the century. These results have stimulated a great amount of interest in counting solutions of equations in groups. This article discusses these results and traces the various developments which these fundamental papers have generated.LetG be a finite group of order |G|. Leto(g) denote the order ofg( G). LetH(s, k)={xG:k|o(x)| sk} wherea/b meansa dividesb and leth(s,k)=|H(s,k)|. Using this notation the simplest of Frobenius' results states ifn/|G|, then/h(n, 1). The minimum value ofh(n, 1) is discussed in the first section. Various conditions are known to insure thath(n, 1)=n. A long standing conjecture of Frobenius states ifn=h(n, 1) thenH(n, 1) is a subgroup (where of coursen/|G|). This conjecture is valid for solvable groups, as well as for various arithmetic conditions.In the second section other divisibility conditions arising from Frobenius' Theorem are discussed. One direction covers more general arithmetic divisibility condition. Another direction has a much wider scope, involving a finite number of equations of an unspecified form and is mainly due to P. Hall. Recently some divisibility conditions involving all groups of a given order have been obtained. Divisibility conditions also hold in infinite groups, and for automorphism analogues of element order. In the next section generalizations to group characters relating back to Frobenius are given. Some of these expressions are used in analyzing properties of group representations and have applications in quantum theory. In the last section clear evidence is established for the combinatorial rather than group-theoretic nature of these results. In particular, some recent work of Snapper links the counting of solutions of equations with the cycle indices in combinatorial theory. Counting solutions of equations in the symmetric groups is also discussed.     

A generalization of Ostrowski's theorem on ultrametric inequalities

Summary We first characterize all the ultrametric functionsf on (assuming both thatf(–x)=f(x) and thatf(x)=0 if and only ifx=0) and then, among these functions, we describe those satisfying the functional equationf(x)·f(x ^–1)=1, for all nonzerox in .Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

Permutation groups of prime degree, a quick proof of Burnside’s theorem

A transitive permutation group of prime degree is doubly transitive or solvable. We give a direct proof of this theorem by Burnside which uses neither S-ring type arguments nor representation theory.Received: 9 December 2004     

A sperner-type theorem

Let P=P ₁×P ₂×...×P _M be the direct product of symmetric chain orders P ₁, P ₂, ..., P _M . Let F be a subset of P containing no l+1 elements which are identical in M–1 components and linearly ordered in the Mth one. Then max |F|cM ^1/2lW(P), where W(P) is the cardinality of the largest level of P, and c is independent of P, M and l. Infinitely many P show that this result is best possible for every M and l apart from the constant factor c.     

A vector-sum theorem in two-dimensional space

Given a finite setX of vectors from the unit ball of the max norm in the twodimensional space whose sum is zero, it is always possible to writeX = {x₁, , x_n} in such a way that the first coordinates of each partial sum lie in [–1, 1] and the second coordinates lie in [–C, C] whereC is a universal constant.     

Characteristic functional equations of polynomials and the morera-carleman theorem

Several characteristic functional equations satisfied by classes of polynomials of bounded degree are examined in connection with certain generalizations of the Morera-Carleman Theorem. Certain functional equations which have nonanalytic polynomial solutions are also considered.     

On the chernoff-savage theorem for dependent sequences

Summary Given a sequence of ϕ-mixing random variables not necessarily stationary, a Chernoff-Savage theorem for two-sample linear rank statistics is proved using the Pyke-Shorack [5] approach based on weak convergence properties of empirical processes in an extended metric. This result is a generalization of Fears and Mehra [4] in that the stationarity is not required and that the condition imposed on the mixing numbers is substantially relaxed. A similar result is shown to hold for strong mixing sequences under slightly stronger conditions on the mixing numbers. Research partially supported by the National Research Council of Canada under Grant No. A-3954.     

A quick distributional way to the prime number theorem

We use distribution theory (generalized functions) to show the prime number theorem. No tauberian results are employed.     

A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.     

Hypergeometric approach to Weideman’s conjecture

By applying the derivative operator to Dixon’s formula, we prove several harmonic number identities including one of the hardest challenge identities conjectured by Weideman (2003). Received: 28 October 2005