Computation with finitely presented groups by Charles C. Sims

By Charles C. Sims

The booklet describes tools for operating with parts, subgroups, and quotient teams of a finitely awarded workforce. the writer emphasizes the relationship with basic algorithms from theoretical laptop technology, rather the speculation of automata and formal languages, from computational quantity idea, and from computational commutative algebra. The LLL lattice aid set of rules and numerous algorithms for Hermite and Smith general types are used to check the Abelian quotients of a finitely provided workforce. The paintings of Baumslag, Cannonito, and Miller on computing non-Abelian polycyclic quotients is defined as a generalization of Buchberger's Gröbner foundation easy methods to correct beliefs within the indispensable workforce ring of a polycyclic team.

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If X = {x1, . . ,Ut =V). Here the equations Ui = V are called defining relations for Q. Under the natural map from X * to Q, the words Ui and V map to the same element. In general, if f : X* -> M is a monoid homomorphism and U and V are words in X* such that f (U) = f (V), then we say that the relation U = V holds in M (relative to f ). If f (U) is the identity element of M, then we normally speak of the relation U = 1, rather than the more consistent U = E. Elements of Mon(X I R) are equivalence classes of words.

Prove that Grp(x I x" = 1) is isomorphic to Z,,. 5. Show that Mon(x, y I xy = yx, x5 = x, y4 = y2, x2y2 = x3y) is finite and determine its order. 6. Find a sequence of Tietze transformations which takes the monoid presentation a2 = b2 = (ab)3 = 1 on generators a and b to the presentation a2 = c3 = (ac)2 = 1 on generators a and c. 7. Let G be a group. Show that G is finitely presented as a group if and only if G is finitely presented as a monoid. 8. Let M = Mon(a, b I ab = 1). Prove that every element of M can be expressed uniquely as [bias].

We just list the elements of the set from first to last. A set with n elements has n! linear orderings, all of which are well-orderings. We shall need some techniques for constructing orderings of infinite sets. Suppose -< is a linear ordering of a set S and n is a positive integer. We can define a linear ordering on the set Sn of n-tuples of elements of S by saying that (sl, ... , sn) -< (t1, . . , Q if and only if there is an integer i with 1 < i < n such that si = tj for 1 < j < i and si - ti.

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