Group theory MOC

Group presentation

A group presentation is a way to present a group as generators and rules relating elements. The idea is to take the free group (i.e. set of words) of the generators, and then use some congruence relation to obtain the desired group properties, i.e. to derive a group from a quotient of a free group. Due to the Correspondence between normal subgroups and congruence relations, the construction can either be done using normal subgroups or congruence relations, but the normal subgroup construction is more convenient.

Let be a set, called the generators, and be a set of words of , called the relators. Let be the Normal closure of in the Free group . Then

Intuitively, is a list of words which should be made equal to the identity. If the congruence relation approach is used instead, a list of relations in the form are given, which can then be converted to the corresponding relators of the form .

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