5 - Coset enumeration

Summary

Chapter 4 dealt with finitely generated subgroups of a free product F of cyclic groups. The basic coset enumeration procedure presented in Sections 4.5 and 4.6 allows us to compute the important-coset automaton A_I(H) for such a subgroup H described by a finite generating set. In general, coset enumeration is a procedure for trying to find A_I(H), where H is a subgroup of F which is finitely generated modulo a normal subgroup N of F. When N is the normal closure of a finite set and H has finite index in F, we can compute A_I(H), and as we shall see in Chapter 6, we can find a finite presentation for H/N.

The notation established in Sections 4.1 to 4.10 remains in effect. Thus X is a finite set and R is a classical niladic rewriting system on X*. The congruence on X* generated by R is ∼, and C is the set of canonical forms for ∼. If U is in X*, then [U] is the ∼-class containing U and Ū is the unique element of [U] ∩ C. If x is in X, then x⁻¹ is the element of X representing the inverse of [x]. If U = x₁ … x_s, then U⁻¹ = x⁻¹_s… x⁻¹₁. If |X| = 1, then F is cyclic of order 2 and we have no trouble studying its subgroups. Therefore we shall assume that |X| ≥ 2.