Since each finite Abelian group and dihedral groups
are determined by their endomorphism monoids in the class of all groups (Lemmas 2.11 and 2.13), we have
Kubo, "The dihedral group
as a family group," in Quantum Field Theory and Beyond, W.
Let now k > 1, and let denote the dihedral group
of order 2k, and let [C.sub.k] = ([C.sub.k], [[less than or equal to].sub.k]) denote the Cambrian lattice associated with [D.sub.k], see .
[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;
(5), we can conclude that N([[GAMMA].sub.1](N))/[+ or -][[GAMMA].sub.1](N) are (,[W.sub.13]) and (,[W.sub.16]), which are isomorphic to the dihedral groups
[D.sub.6], [D.sub.4] for N = 13,16 respectively.
If we accept the notion of shifting perspectives as in McCabe's application of Dihedral Group
Theory, then we need a means of symbolically representing the elements of each perspective and its associated body of knowledge.
For the dihedral group
W = [I.sub.2](m) one easily has [I.sub.2][(m).sup.FC](q) = 1 + 2q + 2[q.sup.2] + ...
Lee  proved that the automorphism group of [G.sub.T](n) is the dihedral group
of symmetries of a regular n-gon.
Let G be the dihedral group
of order [2.sup.m] for a large m and H the (non-normal) subgroup of order 2.
The following lemma, due to Saltman, provides an elegant solution to a local-global embedding problem involving the dihedral group
[D.sub.n] = (a, b | [b.sup.2] = [a.sub.n] = 1, [b.sup.-1] ab = [a.sup.-1])--the dihedral group
of order 2n;
For example, in Figure 5, the q-harmonic polynomials for the dihedral group
G(4, 1, 2) are given by HG[(4,1, 2).sub.q] = [L.sub.0](q).