Milne - Group Theory.. - Free

18 2 FREE GROUPS AND PRESENTATIONS
Proof. Let α be a map X → H. From the universal property of free groups (2.3),
we know that α extends to a homomorphism FX → H, which we again denote α.
Let ιR be the image of R in FX. By assumption ιR ⊂ Ker(α), and therefore the
normal subgroup N generated by ιR is contained in Ker(α). Hence(seep11),α
factors through FX/N = G. This proves the existence, and the uniqueness follows
from the fact that we know the map on a set of generators for X.
Example 2.10. Let G = 〈a, b|a n ,b 2 ,baba〉. We prove that G is isomorphic to D n .
Because the elements σ, τ ∈ D n satisfy these relations, the map
{a, b} →D n , a ↦→ σ, b ↦→ τ
extends uniquely to a homomorphism G → D n . This homomorphism is surjective
because σ and τ generate D n . The relations a n =1, b 2 =1, ba = a n−1 b
imply that each element of G is represented by one of the following elements,
1,...,a n−1 ,b,ab,...,a n−1 b, and so (G : 1) ≤ 2n = (D n : 1). Therefore the homomorphism
is bijective (and these symbols represent distinct elements of G).
Finitely presented groups
A group is said to be finitely presented if it admits a presentation (X, R) withboth
X and R finite.
Example 2.11. Consider a finite group G. LetX = G, andletR be the set of words
{abc −1 | ab = c in G}.
I claim that (X, R) is a presentation of G, andsoG is finitely presented. Let G ′ =
〈X|R〉. Th e map FX → G, a ↦→ a, sends eachelement of R to 1, and therefore
defines a homomorphism G ′ → G, which is obviously surjective. But clearly every
element of G ′ is represented by an element of X, and so the homomorphism is also
injective.
Although it is easy to define a group by a finite presentation, calculating the
properties of the group can be very difficult — note that we are defining the group,
which may be quite small, as the quotient of a huge free group by a huge subgroup.
I list some negative results.
The word problem
Let G be the group defined by a finite presentation (X, R). The word problem for
G asks whether there is an algorithm (decision procedure) for deciding whether a
word on X ′ represents 1 in G. Unfortunately, the answer is negative: Novikov and
Boone showed that there exist finitely presented groups G for which there is no such
algorithm. Of course, there do exist other groups for which there is an algorithm.
The same ideas lead to the following result: there does not exist an algorithm that
will determine for an arbitrary finite presentation whether or not the corresponding