In mathematics, coset enumeration is the problem of counting the cosets 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.[1] 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.[2] More detailed information on correctness, efficiency, and practical implementation can be found in the books by Sims[3] and Holt et al.[4]
^ACE: Advanced Coset Enumerator by George Havas and Colin Ramsay Archived 2007-09-01 at the Wayback Machine
^Rotman, Joseph J. (1995). An Introduction to the Theory of Groups. Springer. ISBN 0-387-94285-8.
^Sims, Charles C. (1994). Computation with Finitely Presented Groups. Cambridge University Press. ISBN 0-521-43213-8.
^Holt, Derek F. (2005). A Handbook of Computational Group Theory. CRC Press. ISBN 1-58488-372-3.
In mathematics, cosetenumeration is the problem of counting the cosets of a subgroup H of a group G given in terms of a presentation. As a by-product...
set of G into disjoint, equal-size subsets called cosets. There are left cosets and right cosets. Cosets (both left and right) have the same number of elements...
{\displaystyle \Gamma } is disconnected and each connected component represents a coset of the subgroup generated by S {\displaystyle S} . If an element s {\displaystyle...
permutation group the Todd–Coxeter algorithm and Knuth–Bendix algorithm for cosetenumeration the product-replacement algorithm for finding random elements of a...
graph of (G, S). The graph is useful to understand cosetenumeration and the Todd–Coxeter algorithm. Coset graphs can be used to form large permutation representations...
work with because they are ill-suited to standard methods such as cosetenumeration. In topology, groups can often be described as finitely presented...
repeated, generating the whole coset S k − 1 τ 1 {\displaystyle S_{k-1}\tau _{1}} , reaching the last permutation in that coset λ k − 1 τ 1 {\displaystyle...
concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics...
n {\displaystyle n} . Then G {\displaystyle G} acts on the set of left cosets of U {\displaystyle U} in G {\displaystyle G} by left shift: g ( h U ) =...
{R} /\mathbb {Q} } of these two groups which is the group formed by the cosets r + Q {\displaystyle r+\mathbb {Q} } of the rational numbers as a subgroup...
left) Each row is a coset with the coset leader in the first column The entry in the i-th row and j-th column is the sum of the i-th coset leader and the j-th...
These were enumerated by length in (Hanusa & Jones 2010). The parabolic subgroups of S ~ n {\displaystyle {\widetilde {S}}_{n}} and their coset representatives...
uncountability of the real numbers Combinatorics Combinatory logic Co-NP Coset Countable countability of a subset of a countable set (to do) Angle of parallelism...
p. 17 Jones, Andrew R. (1996), "A Combinatorial Approach to the Double Cosets of the Symmetric Group with respect to Young Subgroups", Europ. J. Combinatorics...
polyhedron, the polyhedra can be identified with the orbit space G/H – the coset gH corresponds to which polyhedron g sends the chosen polyhedron to. There...
G {\displaystyle e:J\to G} . Instead an enumeration of homomorphisms is used, and since such an enumeration can be constructed uniformly, it results...
in. Modulo a prime, there is only the subgroup of squares and a single coset. The fact that, e.g., modulo 15 the product of the nonresidues 3 and 5,...
of Combinatorics, chapter in a textbook. Arthur T. White, ”Ringing the Cosets,” Amer. Math. Monthly 94 (1987), no. 8, 721-746; Arthur T. White, ”Fabian...
for all u in H the image of u is a linear operator and for all a in the coset G-H the image of a is antilinear (where '*' means complex conjugation):...
elements of the Klein four-group. As with any connected Lie group, the coset spaces of the closed subgroups of the restricted Lorentz group, or homogeneous...
"The Converse of the Intermediate Value Theorem: From Conway to Cantor to Cosets and Beyond" Missouri J. Math. Sci. 26 (2): 134–150 "Large Numbers, Part...
size n. Then the pure state space of Hn can be identified with the compact coset space U ( n ) / ( U ( n − 1 ) × U ( 1 ) ) . {\displaystyle \operatorname...
Global Information