Algebraically closed group
In mathematics, in the realm of group theory, a group is algebraically closed if any finite set of equations and inequations that "make sense" in have a solution in without needing a group extension. This notion will be made precise later in the article in § Formal definition.
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:
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 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.
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 .
This formalizes the idea of substituting elements of for the variables to get true identities and inidentities. In the example the substitutions and yield:
We say the finite set of equations and inequations is consistent with if we can solve them in a "bigger" group . More formally:
The equations and inequations are consistent with if there is a group and an embedding such that the finite set of equations and inequations and has a solution in , where is the unique homomorphism that equals on and is the identity on .
Now we formally define the group to be algebraically closed if every finite set of equations and inequations that has coefficients in and is consistent with has a solution in .
It is difficult to give concrete examples of algebraically closed groups as the following results indicate:
- Every countable group can be embedded in a countable algebraically closed group.
- Every algebraically closed group is simple.
- No algebraically closed group is finitely generated.
- An algebraically closed group cannot be recursively presented.
- A finitely generated group has a solvable word problem if and only if it can be embedded in every algebraically closed group.
The proofs of these results are in general very complex. However, a sketch of the proof that a countable group can be embedded in an algebraically closed group follows.
First we embed in a countable group with the property that every finite set of equations with coefficients in that is consistent in has a solution in as follows:
There are only countably many finite sets of equations and inequations with coefficients in . Fix an enumeration of them. Define groups inductively by:
Now iterate this construction to get a sequence of groups and let:
Then is a countable group containing . It is algebraically closed because any finite set of equations and inequations that is consistent with must have coefficients in some and so must have a solution in .
- A. Macintyre: On algebraically closed groups, ann. of Math, 96, 53-97 (1972)
- B.H. Neumann: A note on algebraically closed groups. J. London Math. Soc. 27, 227-242 (1952)
- B.H. Neumann: The isomorphism problem for algebraically closed groups. In: Word Problems, pp 553–562. Amsterdam: North-Holland 1973
- W.R. Scott: Algebraically closed groups. Proc. Amer. Math. Soc. 2, 118-121 (1951)