[D.sub.n] = (a,b|[b.sup.2] = [a.sup.k] = 1, [b.sup.-1]ab = [a.sup.-1]) - the

dihedral group of order n = 2k;

(i) The

dihedral group [D.sub.8] in order 8: this group has 4 elements of order 4.

This is most transparent in the class of diagrams of highest symmetry, namely, Figure 1, and the diagrams symmetric under the n-th

dihedral group, Figures 2, 3(a), and 4.

The language of [t.sub.b,m] is closed under a group isomorphic to the

dihedral group of order 2m, here denoted [I.sub.2](m).

i) Any infinite tree without leaf is a building of type (W, S) where W is the infinite

dihedral group Z/2Z * Z/2Z and S = {(1, 0), (0, 1)}.

Finally, we consider the Cambrian lattice Ck associated with the

dihedral group [D.sub.k], see for instance [9].

We constructed a finite group and well know group

dihedral group such that the corresponding non-commuting graphs are non isomorphic but the group have the same order.

According to group theory in mathematics, the

dihedral group of order 3 is one of the most robust.

The Group Calculator allows the user to select a group from one of several families: cyclic group of order n,

dihedral group [D.sub.n] of order 2n, the group [Z.sup.*.sub.n] of units modulo n, the abelian group [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the affine group Aff([Z.sub.n]) = {ax + b | a [member of] [Z.sup.*.sub.n] and b [member of] [Z.sub.n]} under composition.

As examples we have [S.sub.3], [A.sub.4], the

dihedral group [D.sub.2m], m odd.

The generators [rho] and [tau] of the

dihedral group [D.sub.n] = {e,[rho],...,[[rho].sup.n- 1],[tau],...,[tau][[rho].sup.n-1]} are such that [[rho].sup.n] = e, [[tau].sup.2] = e and [tau][rho] = [[rho].sup.n- 1][tau].

It is the

dihedral group [D.sub.n] of order 2n with I = (1) = (xb)[.sup.n] = [x.sup.2] = (b)[.sup.2].