4Some group theory
II Algebraic Topology
4.1 Free groups and presentations
Recall that in IA Groups, we defined, say, the dihedral group to be
D
2n
= hr, s | r
n
= s
2
= e, srs = r
−1
i.
What does this expression actually mean? Can we formally assign a meaning to
this expression?
We start with a simple case — the free group. This is, in some sense, the
“freest” group we can have. It is defined in terms of an alphabet and words.
Definition (Alphabet and words). We let
S
=
{s
α
:
α ∈
Λ
}
be our alphabet,
and we have an extra set of symbols
S
−1
=
{s
−1
α
:
α ∈
Λ
}
. We assume that
S ∩ S
−1
= ∅. What do we do with alphabets? We write words with them!
We define
S
∗
to be the set of words over
S ∪ S
−1
, i.e. it contains
n
-tuples
x
1
· · · x
n
for any 0 ≤ n < ∞, where each x
i
∈ S ∪ S
−1
.
Example. Let
S
=
{a, b}
. Then words could be the empty word
∅
, or
a
, or
aba
−1
b
−1
, or
aa
−1
aaaaabbbb
, etc. We are usually lazy and write
aa
−1
aaaaabbbb
as aa
−1
a
5
b
4
.
When we see things like
aa
−1
, we would want to cancel them. This is called
elementary reduction.
Definition (Elementary reduction). An elementary reduction takes a word
us
α
s
−1
α
v and gives uv, or turns us
−1
α
s
α
v into uv.
Since each reduction shortens the word, and the word is finite in length, we
cannot keep reducing for ever. Eventually, we reach a reduced state.
Definition (Reduced word). A word is reduced if it does not admit an elementary
reduction.
Example. ∅, a, aba
−1
b
−1
are reduced words, while aa
−1
aaaaabbbb is not.
Note that there is an inclusion map
S → S
∗
that sends the symbol
s
α
to the
word s
α
.
Definition (Free group). The free group on the set
S
, written
F
(
S
), is the set
of reduced words on S
∗
together with some operations:
(i)
Multiplication is given by concatenation followed by elementary reduction
to get a reduced word. For example, (
aba
−1
b
−1
)
·
(
bab
) =
aba
−1
b
−1
bab
=
ab
2
(ii) The identity is the empty word ∅.
(iii) The inverse of x
1
· · · x
n
is x
−1
n
· · · x
−1
1
, where, of course, (s
−1
α
)
−1
= s
α
.
The elements of S are called the generators of F (S).
Note that we have not showed that multiplication is well-defined — we might
reduce the same word in different ways and reach two different reduced words.
We will show that this is indeed well-defined later, using topology!
Some people like to define the free group in a different way. This is a cleaner
way to define the free group without messing with alphabets and words, but is
(for most people) less intuitive. This definition also does not make it clear that
the free group
F
(
S
) of any set
S
exists. We will state this definition as a lemma.
Lemma. If
G
is a group and
φ
:
S → G
is a set map, then there exists a unique
homomorphism f : F (S) → G such that the following diagram commutes:
F (S)
S G
f
φ
where the arrow not labeled is the natural inclusion map that sends
s
α
(as a
symbol from the alphabet) to s
α
(as a word).
Proof.
Clearly if
f
exists, then
f
must send each
s
α
to
φ
(
s
α
) and
s
−1
α
to
φ
(
s
α
)
−1
.
Then the values of f on all other elements must be determined by
f(x
1
· · · x
n
) = f(x
1
) · · · f (x
n
)
since
f
is a homomorphism. So if
f
exists, it must be unique. So it suffices to
show that this f is a well-defined homomorphism.
This is well-defined if we define
F
(
S
) to be the set of all reduced words, since
each reduced word has a unique representation (since it is defined to be the
representation itself).
To show this is a homomorphism, suppose
x = x
1
· · · x
n
a
1
· · · a
k
, y = a
−1
k
· · · a
−1
1
y
1
· · · y
m
,
where y
1
6= x
−1
n
. Then
xy = x
1
· · · x
n
y
1
· · · y
m
.
Then we can compute
f(x)f(y) =
φ(x
1
) · · · φ(x
n
)φ(a
1
) · · · φ(a
k
)
φ(a
k
)
−1
· · · φ(a
1
)
−1
φ(y
1
) · · · φ(y
m
)
= φ(x
1
) · · · φ(x
n
) · · · φ(y
1
) · · · φ(y
m
)
= f(xy).
So f is a homomorphism.
We call this a “universal property” of
F
(
S
). We can show that
F
(
S
) is the
unique group satisfying the conditions of this lemma (up to isomorphism), by
taking G = F (S) and using the uniqueness properties.
Definition (Presentation of a group). Let
S
be a set, and let
R ⊆ F
(
S
) be any
subset. We denote by
hhRii
the normal closure of
R
, i.e. the smallest normal
subgroup of F (S) containing R. This can be given explicitly by
hhRii =
(
n
Y
i=1
g
i
r
i
g
−1
i
: n ∈ N, r
i
∈ R, g
i
∈ F (S)
)
.
Then we write
hS | Ri = F (S)/hhRii.
This is just the usual notation we have for groups. For example, we can write
D
2n
= hr, s | r
n
, s
2
, srsri.
Again, we can define this with a universal property.
Lemma. If
G
is a group and
φ
:
S → G
is a set map such that
f
(
r
) = 1 for all
r ∈ R
(i.e. if
r
=
s
±1
1
s
±1
2
· · · s
±1
m
, then
φ
(
r
) =
φ
(
s
1
)
±1
φ
(
s
2
)
±1
· · · φ
(
s
m
)
±1
= 1),
then there exists a unique homomorphism
f
:
hS | Ri → G
such that the
following triangle commutes:
hS | Ri
S G
f
φ
Proof is similar to the previous one.
In some sense, this says that all the group
hS | Ri
does is it satisfies the
relations in R, and nothing else.
Example (The
stupid
canonical presentation). Let
G
be a group. We can view
the group as a set, and hence obtain a free group
F
(
G
). There is also an obvious
surjection
G → G
. Then by the universal property of the free group, there is a
surjection
f
:
F
(
G
)
→ G
. Let
R
=
ker f
=
hhRii
. Then
hG | Ri
is a presentation
for G, since the first isomorphism theorem says
G
∼
=
F (G)/ ker f.
This is a really stupid example. For example, even the simplest non-trivial
group
Z/
2 will be written as a quotient of a free group with two generators.
However, this tells us that every group has a presentation.
Example. ha, b | bi
∼
=
hai
∼
=
Z.
Example. With a bit of work, we can show that ha, b | ab
−3
, ba
−2
i = Z/5.