Free group

Cayley graph of a free group formed by two elements a and b

Free group in group theory - group $G$ ${\ displaystyle G}$ $G$ for which there is a subset $S\subset G$ ${\ displaystyle S \ subset G}$ $S \ subset G$ such that each element $G$ ${\ displaystyle G}$ $G$ is written uniquely as the product of a finite number of elements $S$ ${\ displaystyle S}$ $S$ and their reverse . (Uniqueness is understood up to trivial combinations like $st=su^{-1}ut$ ${\ displaystyle st = su ^ {- 1} ut}$ $st = su ^ {{- 1}} ut$ .) They say that $G$ ${\ displaystyle G}$ $G$ (freely) generated $S$ ${\ displaystyle S}$ $S$ and write: $F_{S}$ {\ displaystyle F_ {S}} $F_ {S}$ or $F_{n},$ ${\ displaystyle F_ {n},}$ $F_ {n},$ if a $S$ ${\ displaystyle S}$ $S$ there are many of $n$ ${\ displaystyle n}$ $n$ elements.

A close but distinct concept: a free Abelian group (which is not, generally speaking, a free group).

Content

Constructive definition

It is possible to present an explicit construction of free groups, thereby proving their existence ^[1] ^[2] . We will consider the elements of the set $S$ ${\ displaystyle S}$ “Symbols” and for each symbol $s$ ${\ displaystyle s}$ of $S$ ${\ displaystyle S}$ enter the symbol $s^{-1}$ ${\ displaystyle s ^ {- 1}}$ ; we denote the set of the latter $S^{-1}$ ${\ displaystyle S ^ {- 1}}$ . Let be

T=S\cup S^{-1}

{\ displaystyle T = S \ cup S ^ {- 1}}

.

Define a word over $S$ ${\ displaystyle S}$ as a finite string of (possibly repeating) characters from $T$ ${\ displaystyle T}$ recorded one after another. Together with the operation of concatenation (gluing, attribution) a lot of words over $S$ ${\ displaystyle S}$ becomes a semigroup . We assume that the set of words has an empty word $\varepsilon$ ${\ displaystyle \ varepsilon}$ that does not contain characters. Thus we get the monoid of words over $S .$ ${\ displaystyle S.}$

For example, for $S=\{a,b,c\}$ ${\ displaystyle S = \ {a, b, c \}}$ . $T=\{a,a^{-1},b,b^{-1},c,c^{-1}\}$ ${\ displaystyle T = \ {a, a ^ {- 1}, b, b ^ {- 1}, c, c ^ {- 1} \}}$ , two words:

\alpha =abc^{-1}a,~~\beta =b^{-1}ba^{-1}

{\ displaystyle \ alpha = abc ^ {- 1} a, ~~ \ beta = b ^ {- 1} ba ^ {- 1}}

,

and their concatenation:

\gamma =\alpha \beta =abc^{-1}ab^{-1}ba^{-1}

{\ displaystyle \ gamma = \ alpha \ beta = abc ^ {- 1} ab ^ {- 1} ba ^ {- 1}}

.

For example, $\alpha \varepsilon =\alpha =abc^{-1}a$ ${\ displaystyle \ alpha \ varepsilon = \ alpha = abc ^ {- 1} a}$ .

Next, a word reduction rule is introduced. If in some word after the symbol (symbol) of $S$ ${\ displaystyle S}$ follows (precedes) the corresponding symbol from $S^{-1},$ ${\ displaystyle S ^ {- 1},}$ then deleting this pair of characters is called reduction . A word is called reduced if it can no longer be reduced. Complete reduction is the consistent application of reduction to a given word until it becomes reduced. For example, from the word $\gamma$ ${\ displaystyle \ gamma}$ (see example above) after full reduction, the reduced word is obtained: $abc^{-1}.$ ${\ displaystyle abc ^ {- 1}.}$

Free group $F_{S}$ ${\ displaystyle F_ {S}}$ generated by the set $S$ ${\ displaystyle S}$ (or free group over $S$ ${\ displaystyle S}$ ) is the group of reduced words over $S$ ${\ displaystyle S}$ with a concatenation operation (followed by a complete reduction of the result if necessary).

Properties

All free groups generated by equipotent sets are isomorphic. Moreover, the cardinality of the set that generates a given free group is called its rank .
Free group $F_{n}$ ${\ displaystyle F_ {n}}$ isomorphic to a free product $n$ ${\ displaystyle n}$ copies $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ .
Nielsen-Schreier theorem : any subgroup of a free group is free.
Any group $G$ ${\ displaystyle G}$ there is a quotient group of some free group $F_{S}$ ${\ displaystyle F_ {S}}$ by some of its subgroup H. Behind $S$ ${\ displaystyle S}$ generators can be taken $G$ ${\ displaystyle G}$ . Then there is a natural epimorphism $f:\;F_{S}\to G$ ${\ displaystyle f: \; F_ {S} \ to G}$ . The kernel H of this epimorphism is the set of assignment relations $G=\langle S,H\rangle$ ${\ displaystyle G = \ langle S, H \ rangle}$ .
The commutant of a free group of finite rank has infinite rank. For example, the commutant generated by two elements of a free group $F(a,b)$ ${\ displaystyle F (a, b)}$ Is a free group generated by all switches $[a^{n},b^{m}],\,m,n\neq 0$ ${\ displaystyle [a ^ {n}, b ^ {m}], \, m, n \ neq 0}$ .

Universal property

Free group $F_{S}$ ${\ displaystyle F_ {S}}$ Is in a sense the most general group generated by the set $S .$ ${\ displaystyle S.}$ Namely, for any group $G$ ${\ displaystyle G}$ and any mapping of sets $f\colon S\to G$ ${\ displaystyle f \ colon S \ to G}$ there is a unique group homomorphism $\varphi \colon F_{S}\to G,$ ${\ displaystyle \ varphi \ colon F_ {S} \ to G,}$ for which the following diagram is commutative:

Thus, there is a one-to-one correspondence between the sets of mappings $S\to G$ ${\ displaystyle S \ to G}$ and homomorphisms $F_{S}\to G$ ${\ displaystyle F_ {S} \ to G}$ . For a non-free group, relations in the group would impose restrictions on the possible images of the forming elements of the group.

This property can be taken as the definition of a free group ^[3] , while it is defined only up to isomorphism , like any universal object . This property is called the universality of free groups . Generating set $S$ ${\ displaystyle S}$ called the basis of the group $F_{s}$ ${\ displaystyle F_ {s}}$ . One and the same free group can have different bases.

From the point of view of category theory, a free group is a functor from the category of sets $\mathbf {Set}$ ${\ displaystyle \ mathbf {Set}}$ to group category $\mathbf {Grp}$ ${\ displaystyle \ mathbf {Grp}}$ , which is the left conjugate for a forgetting functor $\mathbf {Grp} \to \mathbf {Set}$ ${\ displaystyle \ mathbf {Grp} \ to \ mathbf {Set}}$ .

Notes

↑ Lindon R., Schupp P. Combinatorial group theory. - M .: Mir, 1980 .-- S. 13.
↑ Ch. 5, § 14 // Fundamentals of group theory / Kargapolov M.I., Merzlyakov Yu.I. - 3rd ed. - M .: Science, 1982. - 288 p.
↑ MacLane S. Categories for the working mathematician = Categories for the working mathematician / Per. from English under the editorship of V.A. Artamonova. - M .: Fizmatlit, 2004 .-- 352 p. - ISBN 5-9221-0400-4 .

Literature

Melnikov O. V., Remeslennikov V. N., Romankov V. A. Chapter II. Groups // General algebra / Under the general. ed. L.A. Skornyakova . - M .: Nauka , 1990. - T. 1. - S. 66-290. - 592 p. - (Reference Mathematical Library). - 30,000 copies. - ISBN 5-02-014426-6 .

[1] Lindon R., Schupp P. Combinatorial group theory. - M .: Mir, 1980 .-- S. 13.

[2] Ch. 5, § 14 // Fundamentals of group theory / Kargapolov M.I., Merzlyakov Yu.I. - 3rd ed. - M .: Science, 1982. - 288 p.

[3] MacLane S. Categories for the working mathematician = Categories for the working mathematician / Per. from English under the editorship of V.A. Artamonova. - M .: Fizmatlit, 2004 .-- 352 p. - ISBN 5-9221-0400-4 .