Group allowing solution of all algebraic equations
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
Suppose we wished to find an element
of a group
satisfying the conditions (equations and inequations):
![{\displaystyle x^{2}=1\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e30f516b22fe867947c871f0f5527e1ce8e40cf)
![{\displaystyle x^{3}=1\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbecde4af22d166fd3699842f0ecc4513825eb6d)
![{\displaystyle x\neq 1\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9749c344d3eea797ddf8fa7a6879c5d5471babca)
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 to the right.
Then the conditions:
![{\displaystyle x^{2}=1\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e30f516b22fe867947c871f0f5527e1ce8e40cf)
![{\displaystyle x\neq 1\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9749c344d3eea797ddf8fa7a6879c5d5471babca)
have a solution in
, namely
.
However the conditions:
![{\displaystyle x^{4}=1\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed9fc8c83958d2e89bc2f94a444b16f176e574e)
![{\displaystyle x^{2}a^{-1}=1\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3eb3f2ea220355e8555d3b6f91a2523158e5f5c)
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 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
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:
![{\displaystyle x_{1}^{2}g_{1}^{4}x_{3}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/480bff1f582198d1522cec4caabfcb2f4f9506c7)
![{\displaystyle x_{3}^{2}g_{2}x_{4}g_{1}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/653d3df4d4c186283a8e2e969138cab295906710)
![{\displaystyle \dots \ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/74f773791864819282d79cbd60c58295d319cc4f)
The set
represents inequations like
![{\displaystyle g_{5}^{-1}x_{3}\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6c36a3ca57c49916877bb073a703e4ebac0696)
![{\displaystyle \dots \ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/74f773791864819282d79cbd60c58295d319cc4f)
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:
![{\displaystyle g_{6}^{2}g_{1}^{4}g_{7}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1748776abae695053fd21eb8067d68ba3f05edd)
![{\displaystyle g_{7}^{2}g_{2}g_{8}g_{1}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e926574a3d693d1899528c99a3cbcd9bd1ece8d)
![{\displaystyle \dots \ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/74f773791864819282d79cbd60c58295d319cc4f)
![{\displaystyle g_{5}^{-1}g_{7}\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0a018490afce9b5ac018e4daae0e629a279c5d)
![{\displaystyle \dots \ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/74f773791864819282d79cbd60c58295d319cc4f)
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 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:
![{\displaystyle D_{0}=C\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1745ccf7375ee0813eab0cb68fac263aa447a2)
![{\displaystyle D_{i+1}=\left\{{\begin{matrix}D_{i}\ &{\mbox{if}}\ S_{i}\ {\mbox{is not consistent with}}\ D_{i}\\\langle D_{i},h_{1},h_{2},\dots ,h_{n}\rangle &{\mbox{if}}\ S_{i}\ {\mbox{has a solution in}}\ H\supseteq D_{i}\ {\mbox{with}}\ x_{j}\mapsto h_{j}\ 1\leq j\leq n\end{matrix}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74574a54c0d112a3c67ba1d567ca4e4a0431a039)
Now let:
![{\displaystyle C_{1}=\cup _{i=0}^{\infty }D_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71fe21421aed447a6300f63a95772626842730a6)
Now iterate this construction to get a sequence of groups
and let:
![{\displaystyle A=\cup _{i=0}^{\infty }C_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fd9495e982e33d7215bcd7d6157732ca051a576)
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
- Algebraic closure
- Algebraically closed field
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)