Editing Coset enumeration

In [[mathematics]], '''coset enumeration''' is the problem of counting the [[coset]]s of a subgroup ''H'' of a [[group (mathematics)|group]] ''G'' given in terms of a [[presentation of a group|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.<ref>[http://www.itee.uq.edu.au/~cram/ce.html ACE: Advanced Coset Enumerator by George Havas and Colin Ramsay] {{webarchive|url=https://web.archive.org/web/20070901213911/http://www.itee.uq.edu.au/~cram/ce.html |date=2007-09-01 }}</ref> The [[Knuth–Bendix algorithm]] also can perform coset enumeration, and unlike the Todd–Coxeter algorithm, it can sometimes solve the [[word problem for groups|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.<ref>{{cite book | first = Joseph J. | last = Rotman | year = 1995 | title = An Introduction to the Theory of Groups | publisher = Springer | isbn = 0-387-94285-8 }}</ref> More detailed information on correctness, efficiency, and practical implementation can be found in the books by Sims<ref>{{cite book | first = Charles C. | last = Sims |authorlink = Charles Sims (mathematician)| year = 1994 | title = Computation with Finitely Presented Groups | url = https://archive.org/details/computationwithf0000sims | url-access = registration | publisher = Cambridge University Press| isbn = 0-521-43213-8 }}</ref> and Holt et al.<ref>{{cite book | first = Derek F. | last = Holt | year = 2005 | title = A Handbook of Computational Group Theory | publisher = CRC Press | isbn = 1-58488-372-3 | url-access = registration | url = https://archive.org/details/handbookofcomput0000holt }}</ref>

==References==
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[[Category:Computational group theory]]