Algebraically closed group

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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 already have a solution in . This idea will be made precise later in the article.

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:

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.

Formal definition of an algebraically closed group[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 .

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)