Presentation of a group

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of some of these generators, and a set R of relations among those generators. We then say G has presentation

Related Topics:
Mathematics - Group - Generators

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:langle S mid R angle.

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Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.

Related Topics:
Isomorphic - Quotient - Free group - Normal subgroup

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As a simple example, the cyclic group of order n has the presentation

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:langle a mid a^n = e angle.

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where e is the group identity.

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Every group has a presentation, and in fact many; a presentation is often the most compact way of describing the structure of the group.

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