# Algorithms for computations of mathematical functions by Luke Y.L.

By Luke Y.L.

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Bq } in N whose images generate NG /G . It follows that N is the normal closure of a finite set; indeed, N = gpG (a1 , . . , a p , b1 , . . , bq ). 2. Every finitely generated metabelian group is finitely presented as a metabelian group. Proof. Let G be a finitely generated group in the class of all metabelian groups. In this class, we can write a presentation G = X; R . It is standard to use “ ” to mean that G ∼ = F(X)/gpF (R), where F(X) is the free metabelian group generated by a finite set X; and gpF (R) is the normal closure of R in F(X).

Observe that Gϕ ≤ B, that B is finitely generated (since Q is), and that B is a semi-direct product of the abelian group A by the finitely generated abelian group Q. 3 Recall that every submodule of a finitely generated ZQ-module is finitely generated as a module (see Chap. 6). 2. Let M be a finitely generated ZQ-module, where Q is a finitely generated abelian group, containing an element t ∈ Q of infinite order in Q. Then there exists a bimonic polynomial f = 1 + c1t + c2t 2 + · · · + t n (ci ∈ Z) such that m f = 0 if and only if m = 0 (m ∈ M).

K). Notice that if m ∈ M1 , then m f1 f2 · · · fk = 0. Put f = 1 + f1 f2 · · · fk . Of course, we need not be concerned with elements m ∈ / M1 . For m ∈ M1 , m f = 0 yields 0 = m f = m + 0 = m, as required. 3 come from commutative algebra. We recommend that the reader consult [1]. 3. Let R be a commutative unital ring. , a multiplicatively closed subset of R containing 1. In addition, assume that 0 ∈ / S, and S contains no zero divisors of R. Then R can be expanded to a commutative unitary ring RS in which the elements of S all have inverses.