Algebraically closed group

In group theory, a group is algebraically closed if any finite set of equations and inequations that are applicable to have a solution in without needing a group extension. This notion will be made precise later in the article in . Suppose we wished to find an element of a group satisfying the conditions (equations and inequations): Then it is easy to see that this is impossible because the first two equations imply . In this case we say the set of conditions are inconsistent with . (In fact this set of conditions are inconsistent with any group whatsoever.) Now suppose is the group with the multiplication table to the right. Then the conditions: have a solution in , namely . However the conditions: Do not have a solution in , as can easily be checked. However if we extend the group to the group with the adjacent multiplication table: Then the conditions have two solutions, namely and . Thus there are three possibilities regarding such conditions: They may be inconsistent with and have no solution in any extension of . They may have a solution in . They may have no solution in but nevertheless have a solution in some extension of . It is reasonable to ask whether there are any groups such that whenever a set of conditions like these have a solution at all, they have a solution in itself? The answer turns out to be "yes", and we call such groups algebraically closed groups. We first need some preliminary ideas. If is a group and is the free group on countably many generators, then by a finite set of equations and inequations with coefficients in we mean a pair of subsets and of the free product of and . This formalizes the notion of a set of equations and inequations consisting of variables and elements of . The set represents equations like: The set represents inequations like By a solution in to this finite set of equations and inequations, we mean a homomorphism , such that for all and for all , where is the unique homomorphism that equals on and is the identity on .