Abstract

The goal of this paper is to start to present a systematic presentation theory for pro-finite groups. Ž . Let G be a fixed finitely-generated pro-finite group and P a free presentation of it; i.e., P 1 R F G 1 Ž . is an exact sequence when F is a finitely generated free pro-finite group. Ž . Ž . We write d F for the minimal number of generators of F and d R for F the minimal number of generators of R as a normal subgroup of F. In G2 Gruenberg presented a systematic study of presentation theory Ž . of a discrete group G mainly for a finite group G and presented three basic problems: Ž . QUESTION 1. Given P 1 R F G 1 for i 1, 2, when F is i i Ž . Ž . a free group, is then d R d R ? F 1 F 2

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