Free group
A group with a system
of generating elements such that any mapping from
into an arbitrary group
can be extended to a homomorphism from
into
. Such a system
is called a system of free generators; its cardinality is called the rank of
. The set
is also called an alphabet. The elements of
are words over the alphabet
, that is, expressions of the form
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where ,
for all
, and also the empty word. A word
is called irreducible if
for every
. The irreducible words are different elements of a free group
, and each word is equal to a unique irreducible word. The number
is called the length of the word
if
is irreducible.
The Nielsen transformations of a finite ordered set of elements of a group are: 1) permutations of two elements of this set; 2) the replacement of one of the
by
; and 3) the replacement of one of the
by
, where
. If a free group
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 is defined analogously to a free group, but within
. It is also called a
-free group, or a relatively-free group (and also a reduced free group). If
is defined by a set of identities
, where
, then a free group of
with a system of generators
is isomorphic to the quotient group
of the free group with system of generators
by the verbal subgroup
defined by
, i.e. the subgroup generated by all elements of
obtained by inserting in words
elements of
. 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
,
,
,
, 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) |
Free group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_group&oldid=18729