View source for Free group

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

<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>

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.

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]]]).

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.

====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>