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can anyone help me out to find some solved example and basic introductory about free product and generators especially the question type like

(a,b,c: a^2=b^2=1 (ab)^3=bc....)

(I just made up this) I have been trying to find some question with easy to understand solution but I could not.

Thanks everyone.

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1 Answer 1

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you should begin with the idea of a free semigroup. you have a set $S$ which can be thought of as a set of symbols $a,b,\dots$. these are called generators. there is no restriction on the cardinality of $S$, it could have only a single member,indeed, in extremis, no members at all -- or it could be an uncountable set. but whatever the cardinality of $S$ we are concerned only with strings of symbols of finite length.

if there is a single generator, say $a$, we can make up only the following strings:

$$ a, aa, aaa, \dots $$ we set a convention to code repetitions with natural number indices, so these strings (words) are represented as $$a, a^2, a^3, \dots$$

for a semigroup with two generators $S=\{a,b\}$ there is a huge increase in complexity. the words which are the elements of the semigroup are all possible strings of finite length. again the index notation is a convenient abbreviation to deal with runs of the same symbol.

the law of composition is to write one word immediately followed by another (catenation). associativity is guaranteed by the simplicity of this operation. but most elements do not commute, for obvious reasons.

the development from semigroup to monoid is accomplished by the simple but profound step of allowing one extra element, the empty string. it is somewhat paradoxical, since it cannot actually be written down! but conceptually it introduces the notion of an identity element. catenation with the identity does not change a word. evidently, like the empty set, the empty string is unique.

the progression from a monoid to a group is essentially performed by an operation on $S$, the set of generators. we represent $S$ as the disjoint union of three subsets, $A,B,C$. the only condition imposed is that $B$ and $C$ have the same cardinality, and that if $B$ is nonempty then we are provided with a bijection $B \rightarrow^{\psi} C$

these sets provide the basis for rewriting rules as follows:

(a) involution: if $x \in A$ then the string $xx$ ($x^2$) may be replaced by the empty string,

(b) inverses: if $y \in B$ and $z \in C$ and $\psi(y) = z$ then the strings $yz$ and $zy$ may be replaced b the empty string wherever they occur.

we may call the rules (a) and (b) the primitive relations which define a free group structure for the free monoid on $S$

the elements of the free group may be regarded as the equivalence classes defined on the free monoid by the primitive relations, though it is psychologically easier to think in terms of maximally reduced words

i hope this brief introduction may provide a useful context for your further study of generators and relations.

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