Finite presentability of Bruck-Reilly extensions of groups

The purpose of this paper is to consider finite generation and finite presentability of a Bruck-Reilly extension S = BR(G, theta) of a group G with respect to an endomorphism theta. It is proved that S is finitely generated if and only if G can be generated by a set of the form U-i=0(infinity) A theta (i), where A subset of or equal to G is finite. The main result states that S is finitely presented if and only if G can be defined by a presentation of the form <A\R > where A is finite and R is of the form U-i=0(infinity) (R) over bar theta (i) finite set of relations (R) over bar. Finally, it is proved that S is finitely presented as an inverse monoid if and only if it is finitely presented as an ordinary monoid. (C) 2001 Academic Press.