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| In [[mathematics]], in the realm of [[group theory]], a [[group (mathematics)|group]] <math>A\ </math> is '''algebraically closed''' if any finite set of equations and inequations that "make sense" in <math>A\ </math> already have a solution in <math>A\ </math>. This idea will be made precise later in the article.
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| ==Informal discussion==
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| Suppose we wished to find an element <math>x\ </math> of a group <math>G\ </math> satisfying the conditions (equations and inequations):
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| ::<math>x^2=1\ </math>
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| ::<math>x^3=1\ </math>
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| ::<math>x\ne 1\ </math>
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| Then it is easy to see that this is impossible because the first two equations imply <math>x=1\ </math>. In this case we say the set of conditions are [[inconsistent]] with <math>G\ </math>. (In fact this set of conditions are inconsistent with any group whatsoever.)
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| {| class="infobox" style="width:auto; font-size:100%"
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| ! style="text-align: center" | <math>G\ </math>
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| |-
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| {| class="wikitable" style="margin: 0"
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| |<math>. \ </math>
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| ! style="background: #ddffdd;"|<math>\underline{1} \ </math>
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| ! style="background: #ddffdd;"|<math>\underline{a} \ </math>
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| |-
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| ! style="background: #ddffdd;"|<math>\underline{1} \ </math>
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| |<math>1 \ </math>
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| |<math>a \ </math>
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| |-
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| ! style="background: #ddffdd;"|<math>\underline{a} \ </math>
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| |<math>a \ </math>
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| |<math>1 \ </math>
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| |}
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| |}
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| Now suppose <math>G\ </math> is the group with the multiplication table:
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| Then the conditions:
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| ::<math>x^2=1\ </math>
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| ::<math>x\ne 1\ </math> | |
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| have a solution in <math>G\ </math>, namely <math>x=a\ </math>.
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| However the conditions:
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| ::<math>x^4=1\ </math>
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| ::<math>x^2a^{-1} = 1\ </math>
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| Do not have a solution in <math>G\ </math>, as can easily be checked.
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| {| class="infobox" style="width:auto; font-size:100%"
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| ! style="text-align: center" | <math>H\ </math>
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| |-
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| {| class="wikitable" style="margin: 0"
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| |<math>. \ </math>
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| ! style="background: #ddffdd;"|<math>\underline{1} \ </math>
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| ! style="background: #ddffdd;"|<math>\underline{a} \ </math>
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| ! style="background: #ddffdd;"|<math>\underline{b} \ </math>
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| ! style="background: #ddffdd;"|<math>\underline{c} \ </math>
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| |-
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| ! style="background: #ddffdd;"|<math>\underline{1} \ </math>
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| |<math>1 \ </math>
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| |<math>a \ </math>
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| |<math>b \ </math>
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| |<math>c \ </math>
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| |-
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| ! style="background: #ddffdd;"|<math>\underline{a} \ </math>
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| |<math>a \ </math>
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| |<math>1 \ </math>
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| |<math>c \ </math>
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| |<math>b \ </math>
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| |-
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| ! style="background: #ddffdd;"|<math>\underline{b} \ </math>
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| |<math>b \ </math>
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| |<math>c \ </math>
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| |<math>a \ </math>
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| |<math>1 \ </math>
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| |-
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| ! style="background: #ddffdd;"|<math>\underline{c} \ </math>
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| |<math>c \ </math>
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| |<math>b \ </math>
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| |<math>1 \ </math>
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| |<math>a \ </math>
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| |}
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| |}
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| However if we extend the group <math>G \ </math> to the group <math>H \ </math> with multiplication table:
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| Then the conditions have two solutions, namely <math>x=b \ </math> and <math>x=c \ </math>.
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| Thus there are three possibilities regarding such conditions:
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| * They may be inconsistent with <math>G \ </math> and have no solution in any extension of <math>G \ </math>.
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| * They may have a solution in <math>G \ </math>.
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| * They may have no solution in <math>G \ </math> but nevertheless have a solution in some extension <math>H \ </math> of <math>G \ </math>.
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| It is reasonable to ask whether there are any groups <math>A \ </math> such that whenever a set of conditions like these have a solution at all, they have a solution in <math>A \ </math> itself? The answer turns out to be "yes", and we call such groups algebraically closed groups.
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| ==Formal definition of an algebraically closed group==
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| We first need some preliminary ideas.
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| If <math>G\ </math> is a group and <math>F\ </math> is the [[free group]] on [[countably]] many generators, then by a '''finite set of equations and inequations with coefficients in''' <math>G\ </math> we mean a pair of subsets <math>E\ </math> and <math>I\ </math> of <math>F\star G</math> the [[free product]] of <math>F\ </math> and <math>G\ </math>.
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| This formalizes the notion of a set of equations and inequations consisting of variables <math>x_i\ </math> and elements <math>g_j\ </math> of <math>G\ </math>. The set <math>E\ </math> represents equations like:
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| ::<math>x_1^2g_1^4x_3=1</math>
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| ::<math>x_3^2g_2x_4g_1=1</math>
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| ::<math>\dots\ </math>
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| The set <math>I\ </math> represents inequations like
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| ::<math>g_5^{-1}x_3\ne 1</math>
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| ::<math>\dots\ </math>
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| By a '''solution''' in <math>G\ </math> to this finite set of equations and inequations, we mean a homomorphism <math>f:F\rightarrow G</math>, such that <math>\tilde{f}(e)=1\ </math> for all <math>e\in E</math> and <math>\tilde{f}(i)\ne 1\ </math> for all <math>i\in I</math>. Where <math>\tilde{f}</math> is the unique homomorphism <math>\tilde{f}:F\star G\rightarrow G</math> that equals <math>f\ </math> on <math>F\ </math> and is the identity on <math>G\ </math>.
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| This formalizes the idea of substituting elements of <math>G\ </math> for the variables to get true identities and inidentities. In the example the substitutions <math>x_1\mapsto g_6, x_3\mapsto g_7</math> and <math>x_4\mapsto g_8</math> yield:
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| ::<math>g_6^2g_1^4g_7=1</math>
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| ::<math>g_7^2g_2g_8g_1=1</math>
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| ::<math>\dots\ </math>
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| ::<math>g_5^{-1}g_7\ne 1</math>
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| ::<math>\dots\ </math>
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| We say the finite set of equations and inequations is '''consistent with''' <math>G\ </math> if we can solve them in a "bigger" group <math>H\ </math>. More formally:
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| The equations and inequations are consistent with <math>G\ </math> if there is a group<math>H\ </math> and an embedding <math>h:G\rightarrow H</math> such that the finite set of equations and inequations <math>\tilde{h}(E)</math> and <math>\tilde{h}(I)</math> has a solution in <math>H\ </math>. Where <math>\tilde{h}</math> is the unique homomorphism <math>\tilde{h}:F\star G\rightarrow F\star H</math> that equals <math>h\ </math> on <math>G\ </math> and is the identity on <math>F\ </math>.
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| Now we formally define the group <math>A\ </math> to be '''algebraically closed''' if every finite set of equations and inequations that has coefficients in <math>A\ </math> and is consistent with <math>A\ </math> has a solution in <math>A\ </math>.
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| ==Known Results==
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| It is difficult to give concrete examples of algebraically closed groups as the following results indicate:
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| * Every [[countable]] group can be embedded in a countable algebraically closed group.
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| * Every algebraically closed group is [[simple group|simple]].
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| * No algebraically closed group is [[Finitely generated group|finitely generated]].
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| * An algebraically closed group cannot be [[presentation of a group|recursively presented]].
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| * A finitely generated group has [[Word problem for groups|solvable word problem]] if and only if it can embedded in every algebraically closed group.
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| The proofs of these results are, in general very complex. However a sketch of the proof that a countable group <math>C\ </math> can be embedded in an algebraically closed group follows.
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| First we embed <math>C\ </math> in a countable group <math>C_1\ </math> with the property that every finite set of equations with coefficients in <math>C\ </math> that is consistent in <math>C_1\ </math> has a solution in <math>C_1\ </math> as follows:
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| There are only countably many finite sets of equations and inequations with coefficients in <math>C\ </math>. Fix an enumeration <math>S_0,S_1,S_2,\dots\ </math> of them. Define groups <math>D_0,D_1,D_2,\dots\ </math> inductively by:
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| ::<math>D_0 = C\ </math> | |
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| ::<math>D_{i+1} =
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| \left\{\begin{matrix}
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| D_i\ &\mbox{if}\ S_i\ \mbox{is not consistent with}\ D_i \\
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| \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\le j\le n
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| \end{matrix}\right.
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| </math>
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| Now let:
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| ::<math>C_1=\cup_{i=0}^{\infty}D_{i}</math>
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| Now iterate this construction to get a sequence of groups <math>C=C_0,C_1,C_2,\dots\ </math> and let:
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| ::<math>A=\cup_{i=0}^{\infty}C_{i}</math>
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| Then <math>A\ </math> is a countable group containing <math>C\ </math>. It is algebraically closed because any finite set of equations and inequations that is consistent with <math>A\ </math> must have coefficients in some <math>C_i\ </math> and so must have a solution in <math>C_{i+1}\ </math>.
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| ==References==
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| * A. Macintyre: On algebraically closed groups, ann. of Math, 96, 53-97 (1972)
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| * B.H. Neumann: A note on algebraically closed groups. J. London Math. Soc. 27, 227-242 (1952)
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| * B.H. Neumann: The isomorphism problem for algebraically closed groups. In: Word Problems, pp 553–562. Amsterdam: North-Holland 1973
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| * W.R. Scott: Algebraically closed groups. Proc. Amer. Math. Soc. 2, 118-121 (1951)
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| [[Category:Properties of groups]]
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