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Thus, for an equation of the fifth degree the various transitive subgroups of the symmetric group of degree five have to be considered.

It is known as the symmetric group of degree n, the only rational functions of the symbols which are unaltered by all possible permutations being the symmetric functions.

Those permutations which leave the product unaltered constitute a group of order n!/2, which is called the alternating group of degree n; it is a self-conjugate subgroup of the symmetric group.

In general, if the equation is given arbitrarily, the group will be the symmetric group.

These processes consist in forming resolvents of the equation corresponding to each distinct type of subgroup of the symmetric group whose degree is that of the equation.