Coset enumeration
Encyclopedia
In mathematics
, coset enumeration is the problem of counting the coset
s of a subgroup H of a group
G given in terms of a presentation
. As a by-product, one obtains a permutation representation for G on the cosets of H. If H has a known finite order, coset enumeration gives the order of G as well.
For small groups it is sometimes possible to perform a coset enumeration by hand. However, for large groups it is time-consuming and error-prone, so it is usually carried out by computer
. Coset enumeration is usually considered to be one of the fundamental problems in computational group theory
.
The original algorithm for coset enumeration was invented by John Arthur Todd and H. S. M. Coxeter. Various improvements to the original Todd–Coxeter algorithm have been suggested, notably the classical strategies of V. Felsch and HLT (Haselgrove, Leech and Trotter). A practical implementation of these strategies with refinements is available at the ACE website. The Knuth–Bendix algorithm also can perform coset enumeration, and unlike the Todd–Coxeter algorithm, it can sometimes solve the word problem
for infinite groups.
The main practical difficulties in producing a coset enumerator are that it is difficult or impossible to predict how much memory or time will be needed to complete the process. If a group is finite, then its coset enumeration must terminate eventually, although it may take arbitrarily long and use an arbitrary amount of memory, even if the group is trivial. Depending on the algorithm used, it may happen that making small changes to the presentation that do not change the group nevertheless have a large impact on the amount of time or memory needed to complete the enumeration. These behaviours are a consequence of the unsolvability of the word problem for groups
.
A gentle introduction to coset enumeration is given in Rotman's text on group theory. More detailed information on correctness, efficiency, and practical implementation can be found in the books by Sims and Holt et al.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, coset enumeration is the problem of counting the coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
s of a subgroup H of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G given in terms of a presentation
Presentation of a group
In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
. As a by-product, one obtains a permutation representation for G on the cosets of H. If H has a known finite order, coset enumeration gives the order of G as well.
For small groups it is sometimes possible to perform a coset enumeration by hand. However, for large groups it is time-consuming and error-prone, so it is usually carried out by computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...
. Coset enumeration is usually considered to be one of the fundamental problems in computational group theory
Computational group theory
In mathematics, computational group theory is the study ofgroups by means of computers. It is concernedwith designing and analysing algorithms anddata structures to compute information about groups...
.
The original algorithm for coset enumeration was invented by John Arthur Todd and H. S. M. Coxeter. Various improvements to the original Todd–Coxeter algorithm have been suggested, notably the classical strategies of V. Felsch and HLT (Haselgrove, Leech and Trotter). A practical implementation of these strategies with refinements is available at the ACE website. The Knuth–Bendix algorithm also can perform coset enumeration, and unlike the Todd–Coxeter algorithm, it can sometimes solve the word problem
Word problem for groups
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element...
for infinite groups.
The main practical difficulties in producing a coset enumerator are that it is difficult or impossible to predict how much memory or time will be needed to complete the process. If a group is finite, then its coset enumeration must terminate eventually, although it may take arbitrarily long and use an arbitrary amount of memory, even if the group is trivial. Depending on the algorithm used, it may happen that making small changes to the presentation that do not change the group nevertheless have a large impact on the amount of time or memory needed to complete the enumeration. These behaviours are a consequence of the unsolvability of the word problem for groups
Word problem for groups
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element...
.
A gentle introduction to coset enumeration is given in Rotman's text on group theory. More detailed information on correctness, efficiency, and practical implementation can be found in the books by Sims and Holt et al.