Algebraically closed group

Summary

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 § Formal definition.

Informal discussion edit

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.

Formal definition edit

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  .

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  .

Known results edit

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   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 let:

 

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  .

See also edit

References edit

  • 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)