For any group G, we write G′ for the commutator subgroup [G, G] and G″for the second commutator subgroup (G′)′ = [G′, G′]. The quotient groups G/G′ and G′/G″ are of course abelian. In this paper we are concerned with the problem of classifying the groups G/G″ when G is finitely generated and the conditions (1) G/G′ is infinite cyclic (2) G/G″ is torsion-free are both satisfied.
(1988). An Overlooked Example of Nonunique Factorization. The American Mathematical Monthly: Vol. 95, No. 4, pp. 339-342.
This chapter discusses the computations in knot theory. The commonest way of presenting a specific knot to the human eye is by a diagram which is to be interpreted as the projection of a curve in 3-dimensional space. There are obviously many ways of coding the information in such a diagram for a computer. To each vertex of the diagram, there correspond two points on the knot, which are referred to as the upper and lower nodes. Each node has a successor arrived at by moving along the knot in the direction indicated by the arrows. Each vertex has one of two possible orientations. If the vertices are then numbered in an arbitrary order, a complete description of the diagram is obtained by listing for each vertex, its orientation, and the successors of its upper and lower nodes.
has a solution in integers. (This and other basic facts stated here without proof can be found in various texts on number theory such as [I ].) An obvious necessary condition for (1) to have a solution is that no factor of d be congruent to 3 modulo 4. The result that this condition is sufficient if d is prime goes back to Legendre; the general problem has been investigated by a number of authors. (See the introduction and list of references in [2].) The theorem we give below does not seem to have been noted explicitly. It is quite elementary and leads directly to various sufficient conditions for the solvability of (1). The theorem and its proof were suggested by the proof given in [1, p. 185] for the case that d is a prime.
#(A, t, ) f f(x, t, ) dz for every Borel set A. The proof occupies Sections 2 and 3. Explicit bounds for the moduli of continuity of f are given by (2.1) and (2.3). This theorem represents a partial extension of results of P. L4vy connected with the notion of "mesure du voisinage" (of the set of zeros of a Brownian motion sample function) introduced and investigated by him [5, 6]. Let F(x, 0) t([--, x], t, ). Then one of Lt!vy’s theorems may be paraphrased as follows: For any fixed , F’() exists with probability one, and is equal to (r/2) times the "measure du voisinage" of the set
