Algebraically closed group
Encyclopedia
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.
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:
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.
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 
.
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:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the realm of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
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
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.) |
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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. |
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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
We first need some preliminary ideas.If
is a group and is the free groupFree group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
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 productFree product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
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
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 simpleSimple groupIn mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
. - No algebraically closed group is finitely generatedFinitely generatedIn mathematics, finitely generated may refer to:* Finitely generated group* Finitely generated monoid* Finitely generated abelian group* Finitely generated module* Finitely generated ideal* Finitely generated algebra* Finitely generated space...
. - An algebraically closed group cannot be recursively presentedPresentation of a groupIn mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
. - A finitely generated group has solvable word problemWord problem for groupsIn mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element...
if and only if it can 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 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 
.
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