4. On the Relations between Gröbner bases and the Subgroup Problem

In this section we want to demonstrate the connection between Gröbner
bases in certain group rings and solutions of the subgroup problem
by rewriting techniques.

Definition 8
Given a subset U of a group ,
let
denote
the subgroup generated by U.
The generalized word problem or subgroup problem is then to
determine, given
,
whether
.

The following theorem links this group theoretic problem to right
respectively left ideals in the respective group ring.

Theorem 2 (see 5.1.2 in [Re95])
Let U be a finite subset of and the group ring
corresponding to .
Further let be a set of polynomials associated to U.
Then the following statements are equivalent: