Free group
|
The Cayley graph of the free group on two generators a and b |
In
mathematics, a
group G is called
free if there is a
subset S of
G such that any element of
G can be written in one and only one way as a product of finitely many elements of
S and their inverses (disregarding trivial variations such as
st-1 =
su-1ut-1).
Note that the notion of free group is different from the notion
free abelian group.
In 1882
Walther Dyck studied the concept of a free group, without naming the concept, in his paper
Gruppentheoretische Studien which was published in the
Mathematische Annalen. The term
free group was introduced by
Jakob Nielsen in 1924.
The group (
Z,+) of
integers is free; we can take
S = {1}. A free group on a two-element subset
S occurs in the proof of the
Banachâ€"Tarski paradox and is described there.
If
S is any set, there always exists a free group on
S. This free group on
S is essentially unique in the following sense: if
F1 and
F2 are two free groups on the set
S, then
F1 and
F2 are
isomorphic, and furthermore there exists precisely one
group isomorphism f :
F1 -> F2 such that
f(
s) =
s for all
s in
S.
This free group on
S is denoted by F(
S) and can be constructed as follows. For every
s in
S, we introduce a new symbol
s-1. We then form the set of all finite
strings consisting of symbols of
S and their inverses. Two such strings are considered equivalent if one arises from the other by replacing two adjacent symbols
ss-1 or
s-1s by the empty string. This generates an
equivalence relation on the set of strings; its
quotient set is defined to be F(
S). Because the equivalence relation is compatible with string concatenation, F(
S) becomes a group with string concatenation as operation.
If
S is the
empty set, then F(
S) is the trivial group consisting only of its
identity element.
The free group on
S is characterized by the following
universal property: if
G is any group and
f :
S â†'
G is any
function, then there exists a unique
group homomorphism T : F(
S) â†'
G such that
T(
s) =
f(
s)
for all
s in
S.
Free groups are thus instances of the more general concept of
free objects in
category theory. Like most universal constructions, they give rise to a pair of
adjoint functors.
Any group
G is isomorphic to a
quotient group of some free group F(
S). If
S can be chosen to be finite here, then
G is called
finitely generated.
If
F is a free group on
S and also on
T, then
S and
T have the same
cardinality. This cardinality is called the
rank of the free group
F.For every cardinal number
k, there is,
up to isomorphism, exactly one free group of rank
k.
If
S has more than one element, then F(
S) is not
abelian, and in fact the
center of F(
S) is trivial (that is, consists only of the identity element).
A free group of finite rank
n > 1 has an
exponential growth rate of order 2
n − 1.
Nielsen-Schreier theorem: Any
subgroup of a free group is free.
A free group of rank
k clearly has subgroups of every rank less than
k.Less obviously, a free group of rank greater than 1 has subgroups of all
countable ranks.
Around 1945,
Alfred Tarski asked whether the free groups on two or more generators have the same
first order theory, and whether this theory is
decidable. Independently, a proof for both problems, and a proof of the first problem, have been announced (both in the affirmative). Neither has yet been judged correct and complete. For details, see the open problems at [
1].
*
Cayley graph*
Generating set of a group*
Presentation of a group*
Free abelian group
