Alperin-Sheriff and Peikert [20] proposed an efficient bootstrapping algorithm by embedding [Z.sub.q] into

permutation group Sq.

The

permutation group isomorphic to [Z.sub.3] contains the permutation matrices 7, X, and [X.sup.2] of the Pauli group, where X is the shift matrix in (1).

Let P = {1, 2, ..., 10} and G = [M.sub.10], [PGL.sub.2](9) or [PGL.sub.2](9) be the primitive

permutation group of degree 10 acting on P.

Symmetry reduction theory based on

permutation group is reported in Section 6.

(3) In the type B process, which is a variant of the carries process, the stationary distribution is proportional to the Macmahon number, which gives us the statistics of the type B-descent of the hyperoctahedral group (signed

permutation group).

In fact, the action of [S.sub.3], the

permutation group in tree elements, preserves the configuration of six points in Fig.

It is a natural extension of (3) from the viewpoint of absolute mathematics, because the symmetric group is interpreted as [S.sub.n] = GL(n,[F.sub.1]), and the generalized

permutation group is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If the right nucleus [N.sub.r](Q) is a normal subgroup, then the equivalence relation induced by the orbits ofthe right-regular

permutation group R([??]) is a normal congruence.

An important class of functions comes from

permutation groups. The orbital profile of a

permutation group G acting on a set E is the function [[theta].sub.G] which counts for each integer n the number, possibly infinite, of orbits of the n-element subsets of E.

A subgroup of [S.sub.X] is called a

permutation group on X.

If the identification group of each structure in !K.sup.P^ coincides with the

permutation group, then all the members of !K.sup.P^ are co-intended.

A hypermap is a pair of permutations ([sigma], [alpha]) on a set {1,2, ..., n}, generating a transitive

permutation group. Its genus g([sigma], [alpha]), a nonnegative integer (see [12]) is defined by