We use the computer constructions of the Monster to determine words in the generators for an element in each conjugacy class, up to algebraic conjugacy. We show how to distinguish the conjugacy classes. You can(not) access the Monster elements database on-line.
to be precise.
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(Preprint and program, tar.gz, 124k.)
LMS J. Comput. Math. 13 (2010) 82–89We use the technique of Fischer matrices to write a program to produce the character table of a group of shape
from the character tables of
G, G:2, 2.G and 2.G:2. We also supply a
simplified proof of a result frequently used in the Fischer matrices
method.
We define a simple, intuitive trajectory-based equivalence relation which accurately captures the relationship between programs and the slices produced by the most commonly used and well-known slicing algorithms, Weiser’s and the PDG algorithm, even for non-terminating programs. Our new equivalence, unlike other approaches, is based on standard semantics. It deals with non-termination by stipulating that a program and its slice must ‘agree’ in all terminating ‘approximations’ based on finite loop unfoldings. We prove that this equivalence is preserved by Weiser’s algorithm and all slicing algorithms based on data and control dependence. We also demonstrate that this new equivalence more accurately captures the behaviour of slicing algorithms than previous work.
Control dependence consists of determining the relationship between nodes of a program’s Control Flow Graph in which one node determines whether or not an- other is executed. This apparently simple relation has proved to be more complex than might appear and has been the subject of recent interest, when it was discov- ered that traditional formulations of control dependence are not suited to reactive programs. There have been many different definitions of control dependence, each defining a different variant of the core notion. This paper defines two purely graph- theoretic semantic notions: weak and strong semantic slices and proves that for each definition of control dependence in the literature, every set closed under control de- pendence induces a graph which is either a weak or a strong semantic slice of the original. These two semantic properties, thus, capture a fundamental property of the transitive closure of all forms of control dependence in slicing. This motivates the introduction of two formulations (and algorithms for their computation) of the transitive closure of control dependence (WCC and SCC) on generalised control flow graphs. We prove that WCC and SCC are the ‘best possible’ in the sense that they give rise to sets which induce minimal weak and strong semantic slices respectively.
