Free Product - Presentation

Presentation

Suppose that

is a presentation for G (where RG is a set of generators and SG is a set of relations), and suppose that

is a presentation for H. Then

That is, G ∗ H is generated by the generators for G together with the generators for H, with relations consisting of the relations from G together with the relations from H (assume here no notational clashes so that these are in fact disjoint unions).

For example, suppose that G is a cyclic group of order 4,

and H is a cyclic group of order 5

Then G ∗ H is the infinite group

Because there are no relations in a free group, the free product of free groups is always a free group. In particular,

where Fn denotes the free group on n generators.

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Famous quotes containing the word presentation:

    “He uses his folly like a stalking-horse, and under the presentation of that he shoots his wit.”
    —William Shakespeare (1564–1616)