John H. Smith

On the representation of−1 as a sum of two squares in an algebraic number field

In a recent paper, P. Chowla proved that −1 can be represented as a sum of two squares in Q(e2πin) if the positive integer n is divisible by a positive integer m ≡ 3(mod 8). In this paper we determine necessary and sufficient conditions in order for −1 to be the sum of two squares in any algebraic number field K. In particular, when K = Q(e2πin) the equation −1 = a2 + b2 is solvable if and only if n is divisible either by 4 or by some odd prime p such that the order of 2(mod p) is even. We show that the set E of such primes consists of all primes ≡ ± 3(mod 8) together with a subset F of the primes ≡ 1(mod 8), where the Dirichlet density of F is 524.

Factoring, into Edge Transpositions of a Tree, Permutations Fixing a Terminal Vertex

If the symmetric group is generated by transpositions corresponding to the edges of a spanning tree we discuss identities they satisfy, including a set of defining relations. We further show that a minimal length factorization of a permutation fixing a terminal vertex does not involve the unique edge incident to that vertex.

Schur complements and LDU decomposition in an additive category

This note establishes the LDU decomposition of a morphism, satisfying the usual sort of hypotheses, in an additive category The Schur complement, defined in terms of this decomposition satisfies a quotient formula. For an appropriate choice of category we get an application to matrices, namely that a set of commutators relative to the blocks of a matrix. A, satisfies the same relations relative to its L D. and U. The Schur complement is shown to describe a quotient or kernel action in the case of an m by n or n by m commutator of rank m linking two matrices, extending some spectral results of Haynsworth. Again a suitable category yields further results on the usual Schur complement. Some non-matrix (topological group) examples are mentioned.

The structure of free prosupersolvable groups

Oltikar [9], and Oltikar and Ribes [lo], have shown that the p-Sylow subgroups of a finitely generated prosupersolvable group are finitely generated. The argument is topological and gives no estimate on the number of generators of the subgroups. In Section 1 we prove a theorem on finite groups which gives such a bound for supersolvable groups. The nature of the bound is such that it holds for prosupersolvable groups via a limit argument. Oltikar 191, also showed, using a cohomologicai argument of Gruenberg (p. 164 of [5]), that the p-Sylow subgroups of a free prosupersolvable group are free pro-p. The argument provides no construction of a set of free generators. In Section 2 we construct, for a given set of primes n, and index set, I, a profinite group, F, and subset, D, indexed by I. The construction is by an iterated semidirect product of free pro-p subgroups, F,, on explicitly given generators and with explicitly given actions of F, on the generators of F, for p > q. We show that D is a set of generators for F and that F is prosupersolvable, which, since the bounds of Section 1 are attained, shows the latter are best possible. In Section 3 we establish a mapping property of F relative to D which shows it is “the” free prosupersolvable n-group on D, thus giving a fairly explicit structure theory, including a Hall system, for the latter object, defined by this mapping property. In particular this provides a noncohomological proof of Oltikar’s second result. In Section 4 we use the Hall system to get similar descriptions (iterated semidirect products of known groups with known actions) of certain canonical subgroups of F. In Section 5 we derive some facts about its automorphism group. For basic facts about profinite groups, in particular the notion of (supernatural) order and p-Sylow subgroups, see [ 11, 131. For the notion of free pro-C product and free pro-C group see [ 1, 3, 5, 7, 141. For extending basic facts about supersolvable groups to prosupersolvable see [9, lo]; for the 256 0021-8693/83

Corrigendum to “Factoring, into edge transpositions of a tree, permutations fixing a terminal vertex” [J. Combin. Theory Ser. A 85 (1) (1999) 92–95]

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Combinatorial congruences fromp-subgroups of the symmetric group

The proof of Theorem 9 of [1] contains an error,1 and the assertion is in fact false, as is shown by the following example. (We use the same symbol for an edge and the associated transposition and multiply from right to left, e.g. (1,2)(1,3) = (1,3,2).) Let the vertices be {1,2,3,4,5,6,7,8} and the edges {a = (1,4),b1 = (3,4),b2 = (2,3), c1 = (4,5), c2 = (5,6), c3 = (6,7), c4 = (7,8)}. Let σ = (1)(2,7)(3,8)(4,6)(5) = ab1c1b2b1c2c1ac3c2c1b1c4c3c2c1b2b1a = b1b2c1b1c2c1b1b2b1c3c2c1b1c4c3c2c1b2b1 (both of length 19). Since, as a permutation of {2,3,4,5,6,7,8} σ has 19 inversions, no product for it involving only b’s and c’s can have smaller length. A bit of (Mathematica-aided) checking confirms that there is no shorter factorization of σ even allowing a, so the example is consistent with the weaker conjecture that there is always some factorization of minimal length not using the edge, i.e. an affirmative answer to the first part of Question 2 of [2] and with Conjecture 1. The example shows that the answer to the second part of Question 2 is (sometimes) affirmative.

A geometric treatment of non-negative generalized inverses

For a finite set,A, we consider subgroups, principallyp-subgroups, of the group,SA, and their action on subsets, families of subsets, and other structures onA. We show that the orders of most orbits are divisible by various powers ofp. These results are applied to obtain congruences involving, for example, binomial and multinomial coefficients, providing alternate proofs for some known results, in some cases generalizations and/or sharpenings, and new congruences involving some other structures.