得到有限可解群及其某些子群的阶与其Fitting 子群的阶之间的若干关系。

In this note we obtain a simple expression of any finote group by means of its generating set. Applying this result we partly solve a conjecture of Cayley graphs proposed by Babai and Seress. We also obtain some other conclusions on diameters on Cayley graphs.

Despite numerous interconnection schemes proposed for distributed multicomputing, systematic studies of classes of interprocessor networks, that offer speed-cost tradeoffs over a wide range, have been few and far in between. A notable exception is the study of Cayley graphs that model a wide array of symmetric networks of theoretical and practical interest. Properties established for all, or for certain subclasses of, Cayley graphs are extremely useful in view of their wide applicability. In this paper, we obtain a number of new relationships between Cayley (di) graphs and their subgraphs and coset graphs with respect to subgroups, focusing in particular on homomorphism between them and on relating their internode distances and diameters. We discuss applications of these results to well-known and useful interconnection networks such as hexagonal and honeycomb meshes as well as certain classes of pruned tori.

An n-node network, with nodes numbered 0 to n-1, is an undirected double-loop network with chord lengths 1 and s (2≤s＜hi2) when each node i (0≤i＜n) is connected to each of the four nodes i±1 and i±s via an undirected link; all node-index expressions are evaluated modulo n. Let n=qs+r, where r (0≤r＜s) is the remainder of dividing n by s. Furthermore, let s=ar+b, where b (0≤b＜r) is the remainder of dividing s by r. In this paper, we provide closed-form formulas for the diameter of a double-loop network for the case q＞r and for a subcase of the case q≤r when b≤aq+1. In the complementary subcase of q≤r, when b＞aq+1, network diameter can be derived by applying the O(log n)-time algorithm of Zerovnik and Pisanski (J.Algorithms, Vol. 14, pp. 226-243, 1993). Obtaining a closed-form formula for diameter of the double-loop network in the latter subcase remains an open problem.

In this note we study the linear symmetry group LS(f) of a Boolean function f of n variables, that is, the set of all £ GLn(2) which leave f invariant, where GLn (2) is the general linear group on the field of two elements. The main problem is that of concrete reproesentation: which subgroups G of GLn(2) can be represented as G=LS(f) for some n-ary k=valued Boolean function f. We call such subgroups linearly representable. The main results of the note may be summarized sa follows: We give a necessary and sufficient condition that a subgroup of GLn(2) is linearly representable and obtain some results on linear representability of its subgroups. Our results generalize some theorems from P.Clote and E. Kranakis [SIAMJ. Comut. 20 (1991) 553-590]; A Kisielewicz [J. Algebra 199 (1998) 379-4031].