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A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f0415501.png" /> with a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f0415502.png" /> of generating elements such that any mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f0415503.png" /> into an arbitrary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f0415504.png" /> can be extended to a homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f0415505.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f0415506.png" />. Such a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f0415507.png" /> is called a system of free generators; its cardinality is called the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f0415508.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f0415509.png" /> is also called an alphabet. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155010.png" /> are words over the alphabet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155011.png" />, that is, expressions of the form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155012.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155014.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155015.png" />, and also the empty word. A word <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155016.png" /> is called irreducible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155017.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155018.png" />. The irreducible words are different elements of a free group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155019.png" />, and each word is equal to a unique irreducible word. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155020.png" /> is called the length of the word <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155021.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155022.png" /> is irreducible.
+
A group  $  F $
 +
with a system  $  X $
 +
of generating elements such that any mapping from  $  X $
 +
into an arbitrary group  $  G $
 +
can be extended to a homomorphism from  $  F $
 +
into  $  G $.  
 +
Such a system  $  X $
 +
is called a system of free generators; its cardinality is called the rank of  $  F $.  
 +
The set  $  X $
 +
is also called an alphabet. The elements of  $  F $
 +
are words over the alphabet  $  X $,
 +
that is, expressions of the form
  
The Nielsen transformations of a finite ordered set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155023.png" /> of a group are: 1) permutations of two elements of this set; 2) the replacement of one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155024.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155025.png" />; and 3) the replacement of one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155026.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155028.png" />. If a free group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155029.png" /> has finite rank, then the Nielsen transformations over the system of free generators lead to new systems of free generators, and any system of free generators can be obtained from any other by successive application of these transformations (Nielsen's theorem, see [[#References|[2]]]). The importance of free groups lies in the fact that every group is isomorphic to a quotient group of a suitable free group. Every subgroup of a free group is also free (the Nielsen–Schreier theorem, see [[#References|[1]]], [[#References|[2]]]).
+
$$
 +
= \
 +
x _ {i _ {1}  } ^ {\epsilon _ {1} } \dots
 +
x _ {i _ {n}  } ^ {\epsilon _ {n} } ,
 +
$$
  
A free group in a [[Variety of groups|variety of groups]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155030.png" /> is defined analogously to a free group, but within <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155031.png" />. It is also called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155033.png" />-free group, or a relatively-free group (and also a reduced free group). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155034.png" /> is defined by a set of identities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155036.png" />, then a free group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155037.png" /> with a system of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155038.png" /> is isomorphic to the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155039.png" /> of the free group with system of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155040.png" /> by the verbal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155041.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155042.png" />, i.e. the subgroup generated by all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155043.png" /> obtained by inserting in words <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155044.png" /> elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155045.png" />. Free groups of certain varieties have special names, for example, free Abelian, free nilpotent, free solvable, free Burnside; they are free groups of the varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041550/f04155049.png" />, respectively.
+
where  $  x _ {i _ {j}  } \in X $,
 +
$  \epsilon _ {j} = \pm  1 $
 +
for all  $  j $,
 +
and also the empty word. A word  $  v $
 +
is called irreducible if  $  x _ {i _ {j}  } ^ {\epsilon _ {j} } \neq x _ {i _ {j + 1 }  } ^ {- \epsilon _ {j + 1 }  } $
 +
for every  $  j = 1 \dots n - 1 $.
 +
The irreducible words are different elements of a free group  $  F $,
 +
and each word is equal to a unique irreducible word. The number  $  n $
 +
is called the length of the word  $  v $
 +
if  $  v $
 +
is irreducible.
 +
 
 +
The Nielsen transformations of a finite ordered set of elements  $  a _ {1} \dots a _ {k} $
 +
of a group are: 1) permutations of two elements of this set; 2) the replacement of one of the  $  a _ {i} $
 +
by  $  a _ {i}  ^ {-} 1 $;
 +
and 3) the replacement of one of the  $  a _ {i} $
 +
by  $  a _ {i} a _ {j} $,
 +
where  $  j \neq i $.
 +
If a free group  $  F $
 +
has finite rank, then the Nielsen transformations over the system of free generators lead to new systems of free generators, and any system of free generators can be obtained from any other by successive application of these transformations (Nielsen's theorem, see [[#References|[2]]]). The importance of free groups lies in the fact that every group is isomorphic to a quotient group of a suitable free group. Every subgroup of a free group is also free (the Nielsen–Schreier theorem, see [[#References|[1]]], [[#References|[2]]]).
 +
 
 +
A free group in a [[Variety of groups|variety of groups]] $  \mathfrak D $
 +
is defined analogously to a free group, but within $  \mathfrak D $.  
 +
It is also called a $  \mathfrak D $-
 +
free group, or a relatively-free group (and also a reduced free group). If $  \mathfrak D $
 +
is defined by a set of identities $  v = 1 $,  
 +
where $  v \in V $,  
 +
then a free group of $  \mathfrak D $
 +
with a system of generators $  X $
 +
is isomorphic to the quotient group $  F/V ( F) $
 +
of the free group with system of generators $  X $
 +
by the verbal subgroup $  V( F) $
 +
defined by $  V $,  
 +
i.e. the subgroup generated by all elements of $  F $
 +
obtained by inserting in words $  v \in V $
 +
elements of $  F $.  
 +
Free groups of certain varieties have special names, for example, free Abelian, free nilpotent, free solvable, free Burnside; they are free groups of the varieties $  \mathfrak A $,  
 +
$  \mathfrak N _ {C} $,  
 +
$  \mathfrak A  ^ {l} $,  
 +
$  \mathfrak B _ {n} $,  
 +
respectively.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Magnus,  A. Karrass,  B. Solitar,  "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Neumann,  "Varieties of groups" , Springer  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Magnus,  A. Karrass,  B. Solitar,  "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Neumann,  "Varieties of groups" , Springer  (1967)</TD></TR></table>

Latest revision as of 19:40, 5 June 2020


A group $ F $ with a system $ X $ of generating elements such that any mapping from $ X $ into an arbitrary group $ G $ can be extended to a homomorphism from $ F $ into $ G $. Such a system $ X $ is called a system of free generators; its cardinality is called the rank of $ F $. The set $ X $ is also called an alphabet. The elements of $ F $ are words over the alphabet $ X $, that is, expressions of the form

$$ v = \ x _ {i _ {1} } ^ {\epsilon _ {1} } \dots x _ {i _ {n} } ^ {\epsilon _ {n} } , $$

where $ x _ {i _ {j} } \in X $, $ \epsilon _ {j} = \pm 1 $ for all $ j $, and also the empty word. A word $ v $ is called irreducible if $ x _ {i _ {j} } ^ {\epsilon _ {j} } \neq x _ {i _ {j + 1 } } ^ {- \epsilon _ {j + 1 } } $ for every $ j = 1 \dots n - 1 $. The irreducible words are different elements of a free group $ F $, and each word is equal to a unique irreducible word. The number $ n $ is called the length of the word $ v $ if $ v $ is irreducible.

The Nielsen transformations of a finite ordered set of elements $ a _ {1} \dots a _ {k} $ of a group are: 1) permutations of two elements of this set; 2) the replacement of one of the $ a _ {i} $ by $ a _ {i} ^ {-} 1 $; and 3) the replacement of one of the $ a _ {i} $ by $ a _ {i} a _ {j} $, where $ j \neq i $. If a free group $ F $ has finite rank, then the Nielsen transformations over the system of free generators lead to new systems of free generators, and any system of free generators can be obtained from any other by successive application of these transformations (Nielsen's theorem, see [2]). The importance of free groups lies in the fact that every group is isomorphic to a quotient group of a suitable free group. Every subgroup of a free group is also free (the Nielsen–Schreier theorem, see [1], [2]).

A free group in a variety of groups $ \mathfrak D $ is defined analogously to a free group, but within $ \mathfrak D $. It is also called a $ \mathfrak D $- free group, or a relatively-free group (and also a reduced free group). If $ \mathfrak D $ is defined by a set of identities $ v = 1 $, where $ v \in V $, then a free group of $ \mathfrak D $ with a system of generators $ X $ is isomorphic to the quotient group $ F/V ( F) $ of the free group with system of generators $ X $ by the verbal subgroup $ V( F) $ defined by $ V $, i.e. the subgroup generated by all elements of $ F $ obtained by inserting in words $ v \in V $ elements of $ F $. Free groups of certain varieties have special names, for example, free Abelian, free nilpotent, free solvable, free Burnside; they are free groups of the varieties $ \mathfrak A $, $ \mathfrak N _ {C} $, $ \mathfrak A ^ {l} $, $ \mathfrak B _ {n} $, respectively.

References

[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[2] W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966)
[3] H. Neumann, "Varieties of groups" , Springer (1967)
How to Cite This Entry:
Free group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_group&oldid=46983
This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article