Symmetric Group - Generators and Relations

Generators and Relations

The symmetric group on n-letters, Sn, may be described as follows. It has generators: and relations:

One thinks of as swapping the i-th and i+1-st position.

Other popular generating sets include the set of transpositions that swap 1 and i for 2 ≤ i ≤ n and a set containing any n-cycle and a 2-cycle of adjacent elements in the n-cycle.

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Famous quotes containing the word relations:

    “It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
    —John Dewey (1859–1952)