mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, coset enumeration is the problem of counting the

coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...

s of a subgroup ''H'' of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

''G'' given in terms of a

presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...

. As a by-product, one obtains a

permutation representation In mathematics, the term permutation representation of a (typically finite) group G can refer to either of two closely related notions: a representation of G as a group of permutations, or as a group of permutation matrices. The term also refers ...

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 A computer is a machine that can be Computer programming, programmed to automatically Execution (computing), carry out sequences of arithmetic or logical operations (''computation''). Modern digital electronic computers can perform generic set ...

. Coset enumeration is usually considered to be one of the fundamental problems in

computational group theory In mathematics, computational group theory is the study of group (mathematics), groups by means of computers. It is concerned with designing and analysing algorithms and data structures to compute information about groups. The subject has attracte ...

. 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 word is a basic element of language that carries meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no consensus among linguists on its ...

. 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.

References

{{DEFAULTSORT:Coset Enumeration Computational group theory