Algebraically closed group

In group theory, a group formula_1 is algebraically closed if any finite set of equations and inequations that "make sense" in formula_1 have a solution in formula_1 without needing a group extension. This notion will be made precise later in the article in .
Informal discussion.
Suppose we wished to find an element formula_4 of a group formula_5 satisfying the conditions (equations and inequations):
Then it is easy to see that this is impossible because the first two equations imply formula_9. In this case we say the set of conditions are inconsistent with formula_5. (In fact this set of conditions are inconsistent with any group whatsoever.)
Now suppose formula_5 is the group with the multiplication table:
Then the conditions:
have a solution in formula_5, namely formula_15.
However the conditions:
Do not have a solution in formula_5, as can easily be checked.
However if we extend the group formula_19 to the group formula_20 with multiplication table:
Then the conditions have two solutions, namely formula_21 and formula_22.
Thus there are three possibilities regarding such conditions:
It is reasonable to ask whether there are any groups formula_29 such that whenever a set of conditions like these have a solution at all, they have a solution in formula_29 itself? The answer turns out to be "yes", and we call such groups algebraically closed groups.
Formal definition.
We first need some preliminary ideas.
If formula_5 is a group and formula_32 is the free group on countably many generators, then by a finite set of equations and inequations with coefficients in formula_5 we mean a pair of subsets formula_34 and formula_35 of formula_36 the free product of formula_32 and formula_5.
This formalizes the notion of a set of equations and inequations consisting of variables formula_39 and elements formula_40 of formula_5. The set formula_34 represents equations like: 
The set formula_35 represents inequations like
By a solution in formula_5 to this finite set of equations and inequations, we mean a homomorphism formula_50, such that formula_51 for all formula_52 and formula_53 for all formula_54, where formula_55 is the unique homomorphism formula_56 that equals formula_57 on formula_32 and is the identity on formula_5.
This formalizes the idea of substituting elements of formula_5 for the variables to get true identities and inidentities. In the example the substitutions formula_61 and formula_62 yield:
We say the finite set of equations and inequations is consistent with formula_5 if we can solve them in a "bigger" group formula_69. More formally:
The equations and inequations are consistent with formula_5 if there is a groupformula_69 and an embedding formula_72 such that the finite set of equations and inequations formula_73 and formula_74 has a solution in formula_69, where formula_76 is the unique homomorphism formula_77 that equals formula_78 on formula_5 and is the identity on formula_32.
Now we formally define the group formula_1 to be algebraically closed if every finite set of equations and inequations that has coefficients in formula_1 and is consistent with formula_1 has a solution in formula_1.
Known Results.
It is difficult to give concrete examples of algebraically closed groups as the following results indicate:
The proofs of these results are in general very complex. However, a sketch of the proof that a countable group formula_85 can be embedded in an algebraically closed group follows.
First we embed formula_85 in a countable group formula_87 with the property that every finite set of equations with coefficients in formula_85 that is consistent in formula_87 has a solution in formula_87 as follows:
There are only countably many finite sets of equations and inequations with coefficients in formula_85. Fix an enumeration formula_92 of them. Define groups formula_93 inductively by:
Now let:
Now iterate this construction to get a sequence of groups formula_97 and let:
Then formula_1 is a countable group containing formula_85. It is algebraically closed because any finite set of equations and inequations that is consistent with formula_1 must have coefficients in some formula_102 and so must have a solution in formula_103.