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Wikipedia / Algebraically closed group

In group theory, a group is algebraically closed if any finite set without needing a group extension. This notion will be made precise later in the article in .

Informal discussion

satisfying the conditions (equations and inequations):

. In this case we say the set of conditions are inconsistent. (In fact this set of conditions are inconsistent with any group whatsoever.) is the group with the multiplication table to the right.

Then the conditions:

.

However the conditions:

, as can easily be checked. with the adjacent multiplication table:

.

Thus there are three possibilities regarding such conditions:

  • .

  • .

  • .

itself? The answer turns out to be "yes", and we call such groups algebraically closed groups.

Formal definition

We first need some preliminary ideas.

is the free group on countably many generators, then by a finite set of equations and inequations with coefficients in the free product.

represents equations like:

represents inequations like

By a solution.

yield:

We say the finite set of equations and inequations is consistent with. More formally:

.

to be algebraically closed.

Known results

It is difficult to give concrete examples of algebraically closed groups as the following results indicate:

can be embedded in an algebraically closed group follows.

as follows:

inductively by:

Now let:

and let:

.

See also

Algebraic closure

References

  • 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)