To the methods of deriving discrete groups of W_p-symmetry
by
A.P. Lungu
State University of Moldova

The methods of deriving the groups of P-symmetry of different types are based on homomorphic mapping and its properties [1]. The solution of analogous problems for W_p-symmetry [2] demands the generalization of homomorphisms as the natural left quasihomomorphism [3]. Moreover, it requires the investigation of some their properties.

The groups G^(W_p) of W_p-symmetry are subgroups of the left standard Cartesian wreath product of initial group P of permutations with discrete group G of classical symmetry as their generating group:

G(Wp) <= G \smallint   P = G \rightthreetimes W.

Let us have groups G and P and a homomorphism j:G --> Aut P. The mapping \mu of the group G onto the subset P' of the group P by the rule \mu(g)=p is called a left quasihomomorphism if for any g_i and g_j from G

\mu(gigj)=[\mu(gi)] \leftharpoonup jgi   \mu(gj) = (pi) \leftharpoonup jgi   pj=pk,

where p_i, p_j, p_k in P' and [ \leftharpoonup || (j_gi)] = j(g_i). When Ker j=1, the mapping \mu is called a natural left quasihomomorphism.

At the left quasihomomorphism \mu in general the image of G \mu(G)=P' subset P is not a group, but P' always contains the unit of the group P. Ker\mu=H is a subgroup in G; the index of this subgroup coincides with the power of \mu(G).

The natural left quasihomomorphism \mu of the group G into the group W=[`(\prod)]_{gi in G}P^g_i under which the automorphism [ \leftharpoonup || (j_g)] \equiv [ \leftharpoonup || g] operates on the elementes w of \mu(G) by means of left g-translations of their components is called an exact natural left quasihomomorphism.

The mapping [(\mu)\tilde] of the group G onto the subset X of the set of all left cosets of group W by its subgroup V is called a generalized exact natural left quasihomomorphism if for any g_i and g_j from G conditions [(\mu)\tilde] (g_i)=w_iV and [(\mu)\tilde] (g_j)=w_jV imply [(\mu)\tilde] (g_ig_j)=(w_iV)^g_j*w_jV=w_kV, where w_iV, w_jV, w_kV in X. Note that in the case of V=w_o the mapping [(\mu)\tilde] is an ordinary exact natural left quasihomomorphism.

Any group G^(W_p) of W_p-symmetry with the finite group W can be derived from its finite generating group G and group W by the following steps: 1) to find in W all subgroups V and subsets W', which are decomposed in left cosets by its subgroup V, and in G all proper subgroups H with the index equal to the power of set of all left cosets of W' by V and for which there is the isomorphism \lambda of factor-groups        G₁/H and W₁/V₁ (\lambda: G₁/H --> W₁/V by the rule \lambda(Hg)=wV), where G₁ <= G, W₁ <= DiagW and V₁=V \cap Diag W <= W₁; 2) to construct a generalized exact natural left quasihomomorphism [(\mu)\tilde] of the group G onto the set of all left cosets of W' by the subgroup V by the rule [(\mu)\tilde] (Hg)=wV and which preserves the correspondence between the elements of factor-groups G₁/H and W₁/V₁ received as the result of isomorphism \lambda; 3) to combine pairwise each g' of Hg with each w' of wV=[(\mu)\tilde] (g'); 4) to introduce into the set of all these pairs the operation

w_ig_i * w_jg_j = w_i^g_j[ \rightharpoonup || (\tau_gi)](w_j)g_ig_j=w_kg_k.

References

[1]. Zamorzaev A.M., Karpova Yu. S., Lungu A.P. and Palistrant A.F., P-symmetry and its Further Development. Shtiintsa: Kishinev, 1986 (in Russian).

[2]. Lungu A.P., The method of deriving semijunior and pseudojunior groups of W-symmetry. Bul. AS a RM. Matematica, 1994, no. 2(15), p.29-39 (in Russian).

[3]. Lungu A.P., Some properties of left quasihomomorphis mapping. Bul. AS a RM. Mat-ca, 1995, no. 1(17) p. 78-81 (in Russian).

Date received: February 9, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadw-19.