Free Group
Free Group
Let
be a subset of a group . We say that is a free group with basis if it satisfies the following universal property: for every group and every function there exists a unique homomorphism with for every . 
In other words, it means that a homomorphism
Proposition
A finite group is never free.
Proof This is straightforward.
Alphabets and Words
Monoid
A set
is a monoid if all the group axioms hold except for the inverse. A set is a semigroup if all the monoid axioms hold except for the identity.
Word
Let
be an arbitrary set. A word (string) with letters in is an finite expression of the form etc. with . We denote the empty word by .
Proposition
The set of words
is a monoid under concatenation
Cancellation Rule and Reduced Word
Consider
. That is, we add formal symbols for inverses. We consider now the set of words . The cancellation law is introduced as if or occur in , then we replace it with the empty word. A word is called reduced if no more cancellations are possible.
Proposition
There is only one reduced form for any given word
.
Word Equivalence
We call two words
and equivalent if they have the same reduced form. Write .
Proposition
The relation
on is an equivalence relation. Moreover, it respects the composition in ,that is, if and then .
Proof The proof is straightforward.
Theorem
The set
of equivalence classes in with the induced law of composition is a group.
Prop 
Proposition
Every group is a quotient of a free group. Indeed, define
sending the alphabet to the group elements .
Relations
Relation in Group
A relation among elements
in a group is a word in the free group on such that in . e.g. In the dihedral group we have elements (rotation) and (reflection) and they satisfies
Lemma
Let
be a subset in a group . There exists a unique minimal normal subgroup of that contains . The elements of have the explicit form
Normal Closure
The above normal subgroup
is called the normal closure of in .
Generated Group
Let
be the free group on the set and let . The group generated by with relation is the quotient where is the normal closure of in . It is denoted .
e.g.