Free group

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)