Abstract. In this article we analyze the essence of the recent "physical'' generalizations of classical symmetry from the viewpoint of the split extensions of groups. The principal results of the general theory of groups of W_p- and W_q-symmetry are given. The methods of deriving generalized symmetry groups of different types are formulated. All these methods are based on right, left and crossed quasi-homomorphisms and their generalizations. The principal properties of diverse quasi-homomorphic mappings are examined. The article is illustrated with many examples of colored and ïndexed" geometrical figures that are described by the groups of P-,`P-, W_p- or W_q-symmetry.

Keywords: color symmetry groups; W-symmetry groups; action of automorphism; generalizations of homomorphism; quasi-homomorphic reflections; wreath products.
 

AMS Subject Classification:

Primary: 20H15; 20E36; 20B25; 20A05; 51M20; 51N30;

Secondary: 52; 82.
 
 

It is well known that the modern theory of symmetry of the real crystal gives rise to new generalizations.

One of the essential generalizations of classical symmetry is the P-symmetry of A.M.Zamorzaev [1-3]. In the case of P-symmetry, the transformations of the qualities attributed to the points, are combined directly with the geometrical transformations and do not depend on the choice of points. Other proposed generalizations such as polychromatic symmetry of Wittke-Garrido [4] or complex [5] symmetry are not included in the scheme of A.M.Zamorzaev's P-symmetry. For these generalizations, the transformations of the qualities attributed to the points, essentially depend on the choice of points. The mentioned generalizations are included in W_p-symmetry that was introduced by V.A.Koptsik and I.N.Kotsev [6] and was developed in the cycle of papers [3, 6-21, 28, 33, 35].

On the other hand, V.A.Koptsik in [22] proposed the notion of Q-symmetry. The essence of Q-symmetry consists actually in the following: the transformation of Q-symmetry g^(q) is composed from the components g and q, where g is transformation of symmetry which operates both on points (the atoms of crystal) and on indexes (vector or tensor) by the given rule independent of the points, and q is a supplementary transformation of these indexes. Uniting problems to be solved for W_p-symmetry and Q-symmetry (`P-symmetry [23-25]), we shall get a generalization named W_q-symmetry [26-33].

The methods of deriving the groups of P-symmetry of different types [1, 2] are based on homomorphic mapping and its properties. The solution of analogous problems for`P]- and W-symmetry demands the generalization of homomorphisms as the right quasi-homomorphism [23, 3], the natural left quasi-homomorphism [14], and the crossed quasi-homomorphism [27]. Moreover, it requires the investigation of some their properties.
 
 

Let us have groups G and P. Construct the Cartesian product W of isomorphic copies of the group P which are indexed by elements of G: W =`Õ<sub>gi Î G</sub>P<sup>g<sub>i</sub></sup>, where P<sup>g<sub>i</sub></sup> @ P. Moreover, construct the isomorphic injection f: G®Aut W by the rule f(g) =  (where the automorphism  makes the left g-translation of the components in w ÎW, i.e. : w®w<sup>g</sup>), and also the homomorphism t: G ® F £ Aut W (where t(g) =  and (w) = gwg<sup>-1</sup>). The set G<sup>*</sup> of pairs wg (where w ÎW and gÎG) forms a group with the operation w<sub>i</sub>g<sub>i</sub>* w<sub>j</sub>g<sub>j</sub> = w<sub>k</sub>g<sub>k</sub>, where g<sub>k</sub> = g<sub>i</sub>g<sub>j</sub> and w<sub>k</sub> = w<sub>i</sub><sup>g<sub>j</sub></sup>(w<sub>j</sub>). We call G<sup>*</sup> the crossed standard Cartesian wreath product of groups P and G, accompanied with the homomorphism t : G ®Aut W by the rule t(g<sub>i</sub>) = , and denote it by the symbol P G`[26, 28, 31].

If W is a direct product of the isomorphic copies of the group P which are indexed by elements of G, then at analogous mode it defined the crossed standard direct wreath product of groups P and G.
 
 

Ascribe to each point of a geometrical figure F with the discrete symmetry group G at least one index, which means a non-geometrical feature, from the set N = {1,2,...,m}, and fix a certain transitive group P of the permutations of these indexes.

The transformation of P-symmetry is defined to be an isometric mapping g^(p) = gp = pg of the "indexed" geometrical figure F^(N) onto itself in which the geometrical component g operates only on points, and the indexes are transformed by the permutation p of the group P. The set G^(P) of transformations of P-symmetry of any "indexed'' geometrical figure F^(N) forms a group with the operation
 

where g_k = g_i g_j and p_k = p_i p_j [1]. The groups G^(P) of P-symmetry are subgroups of the direct products of the group P of permutations with their generating discrete groups G of classical symmetry.

Example 1. Let F be a geometrical figure with the classical symmetry group G = 4 = (1, 4, 2, 4^-1) (we use Shubnikov's symbols [34]) and N = { 1,2,3,4 }, where the yellow, green, red and blue colors are denoted with the indexes 1,2,3 and 4, respectively. Moreover, P = (e, p₁ = (1234), p₂ = (13)(24), p₃ = (1432)). Then g^(p) = gp = 4p₁ is a transformation of P-symmetry of the colored square represented in Figure 1.
 
 

The group of above P-symmetry is
 

with the group/subgroup symbol 4/1, where 4 is the symbol of the generating group and 1 is the symbol of its classical symmetry subgroup [2].

The transformation of `P-symmetry is defined to be an isometric mapping g^(p) = pg of the "indexed'' geometrical figure F^(N) onto itself in which the geometrical component g operates both on points and on indexes by the given rule independent of the points, but the permutation p is only a compensating permutation of indexes to map F^(N) onto itself and p Î P. In this case the components p and g of the transformation g^(p) in general do not commute: pg ¹gp, that is why p ¹ gpg^-1.

The set G<sup>(`P)</sup> of transformations of `P-symmetry of any "indexed'' geometrical figure F^(N) forms a group with the operation

where g_k = g_i g_j, p_k = p_i(p_j) and (p_j) = g_i p_j g_i^-1 = p_sÎP. The groups of `P-symmetry are subgroups of the right semi-direct product of the group P with group G, accompanied with the homomorphism f: G®Aut P by the rule f(g_i) = [23, 3].

Example 2. Let us have six different positions of vectors with the equal modules, that are numbered using 1,2,3,4,5 and 6 in accordance with the scheme represented in Figure 2.
 

In this case, the "ïndexed" figures represented in Figure 3a) and Figure 3b) are described from the viewpoint of P-symmetry by the same group with the group/subgroup symbol 4/1.
 
 

From the viewpoint of `P-symmetry the "ïndexed" figures represented in Figures 3a) and 3b) are described by the different groups [3].

Construct the Cartesian product W of isomorphic copies of the group P which are indexed by elements of G, i.e. W = `Õ_{giÎ G}P^g_i, where P^g_i @ P.
The transformation of W_p-symmetry is defined to be an isometric mapping g^(w) = gw of the "ïndexed" geometrical figure F^(N) onto itself in which the geometrical component g operates only on points M_k = g_k(M₁) of the figure F^(N) (where M₁ is a point of general position of the figure F with respect to the group G), not affecting indexes, and the indexes ascribed to the point M_k are transformed by the permutation p^g_k which is the "g_k-component'' in w. The set of transformations of W_p-symmetry of the given "indexed'' geometrical figure F^(N) forms a group G^(W_p) with the operation

Example 3. Let's have the groups G = 4 and P = (e, p = (12)), where the yellow and red colors are denoted with indexes 1 and 2. Then g^(w) = gw = 4 < p¹, e⁴, p², e^4-1> is a transformation of W_p-symmetry of the colored square represented in Figure 4.

We note that in the case when G = 4 and P = (e, p = (12)), the transformation of W_p-symmetry g^(w) = 4 < p¹, p⁴, p², p^4-1> exists too, which may be formally considered as a transformation of P-symmetry (it satisfies the conditions of the respective definition - there is one rule of the transformation of colors for all equivalent points) [35]. This transformation and the transformation of P-symmetry g^(p) = gp = 4p describe the same colored square (Figure 5).

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

G(Wp) £ G P = GW.

The transformation of W_q-symmetry is defined to be an isometric mapping g^(w) = wg of the "indexed'' geometrical figure F^(N) onto itself in which the geometrical component g operates both on points M_k = g_k(M₁) of the figure F^(N) (where M₁ is a point of general position of the figure F with respect to the group G) and on indexes by the given rule independent of the points, but the permutation p^g_k ("g_k-component'' in w) is only a compensating permutation of indexes in the point M_k to map F^(N) onto itself. The set G^(W_q) of transformations of W_q-symmetry of the given "indexed'' figure F^(N) forms a group with the operation

where g_k = g_i g_j, w_k = w_i^g_j(w_j), w_i^g_j(g_s) = w_i(g_jg_s) and (w_j) = g_i w_j g_i^-1 [26].

Example 4. Let us have four different positions of vectors with the equal modules that are numbered using 1,2,3 and 4 in accordance with the scheme that is represented in Figure 6;
 
 

let's have the groups G = C₄ = 4 = (1, 4, 2, 4^-1) and P = (e, p₁ = (1234), p₂ = (13)(24), p₃ = (1432)) @C₄. Then g^(w) = wg = < p₁¹, p₃⁴, p₁², p₃^4-1> 4 is a transformation of W_q-symmetry of the "ïndexed" figure represented in Figure 7.

We note that in the case when G = C_4v = 4 ·m = (1, 4, 2, 4^-1, m₁, m₂, m₃, m₄) and P@C₄, the transformation of W_q-symmetry g^(w) = wg = < p₂¹, p₂⁴, p₂², p₂^4-1, p₂^m₁, p₂^m₂, p₂^m₃, p₂^m₄ > 4 exists too, which may be considered formally as a transformation of `P-symmetry. This transformation and the transformation of `P-symmetry g^(p) = pg = p₂4 describe the same "ïndexed" figure F^(N) represented in Figure 8.

The groups G^(W_q) of W_q-symmetry are subgroups of the crossed standard Cartesian wreath product of the initial group P of permutations and the discrete group G of classical symmetry (as their generating group), accompanied with the homomorphism t: G®Aut W by the rule t(g_i) =:

G(Wq)£P G = W G.

Let G^(W) be a group of W_p- or W_q-symmetry with the initial group P, generating group G and subset W^¢ = {w | g^(w)ÎG^(W)} ÍW. Identifying the groups G and W with their isomorphic injections into GW = G P (into WG = PG, respectively) by the rules g ®gw₀, where w₀ is the unit of the group W, and w 1w, where 1 is the unit of the group G (g w₀g and ww1, respectively), we find the symmetry subgroup H = G^(W)Ç G and the subgroup V = G^(W)Ç W = G^(W)Ç W^¢ of W-identical transformations of the group G^(W).

The group G^(W) is called senior, junior or V-middle if w₀ < V = W^¢ = W, w₀ = V < W^¢ = W or w₀ < V < W^¢ = W, respectively. If W^¢ is a non-trivial subgroup of W, then the group G^(W) is called W^¢-semi-senior, W^¢-semi-junior or (W^¢, V)-semi-middle according to the cases when w₀ < V = W^¢, w₀ = V < W^¢ or w₀ < V < W^¢. If W^¢ÌW, but W^¢ is not a group, the group G^(W) is called W^¢-pseudo-junior or (W^¢, V)-pseudo-middle when w₀ = V Ì W^¢ or w₀ < V Ì W^¢.

Let G^(W_p) be a group of W_p-symmetry with the initial group P, generating group G, subset W^¢ = {w | g^(w) ÎG^(W_p)}, symmetry subgroup H and the subgroup V of W-identical transformations. Then: 

        1) the mapping f of the group G^(W_p) onto the group G by the rule f[g^(w)] = g is homomorphic with the kernel V; 

        2) the group G^(W_p) contains as its subgroup the group G₁^(W₁) of P-symmetry (which is determined by initial group P of permutations, where W₁£ Diag W @P and W₁ÌW^¢) from the family with the generating group G₁ (G₁ £ G), with the symmetry subgroup H (where H G₁ but H ¹ G) and with the subgroup V₁ of W-identical transformations (where V₁ = V Ç Diag W).

Moreover, if W is a finite group then: 

        1) V^g = wVw^-1, where g and w are components of the transformation g^(w) from G^(W_p); 

        2) all the elements of a right coset Hg of the group G by H are combined in pairs only with the elements of one left coset wV, and the elements of different cosets Hg_i and Hg_j with the elements of different cosets w_iV and w_jV [18-20, 28].

Let G^(W_q) be a group of W_q-symmetry with the generating group G, permutation group P, (i.e. W = `Õ_{gi Î G}P^g_i, where P^g_i @ P), subset W^¢ = {w | g^(w) Î G^(W_q)}, with the kernel H₁ of accompanying homomorphism t: G ® Aut W, symmetry subgroup H and the subgroup V of W-identical transformations. In this case the following conditions are satisfied: 

        1) the mapping f of the group G^(W_q) onto the generating group G by the rule f[g^(w)] = g is homomorphic with the kernel V; 

        2) the group G^(W_q) contains as subgroup the group H₁^(W_p) of W_p-symmetry (which is determined by the initial group P of permutations) from the family with the generating group H₁, with the symmetry subgroup H^¢ (where H^¢ = H Ç H₁) and with the same subgroup V of W-identical transformations; 

        3) the group G^(W_q) contains as a subgroup the group G₁^(W₁) of `P-symmetry (which is determined by initial group P of permutations) from the family with the generating group G₁ (where G₁ £ G), with the same kernel H₁ of the accompanying homomorphism and with the set W₁ = {w | g^(w) Î G₁^(W₁)} of multi-component permutations, where W₁ = W^¢ Ç Diag W.

If W is a finite group of multi-component permutations, then: 

        1) V^g = V for any transformation g^(w) from G^(W_q), i.e. V is -invariant subgroup; 

        2) (V) = (w_i^(g_j))^-1Vw_i^g_j for transformations g_i^(w_i) and g_j^(w_j) from G^(W_q) [26-28].
 
 

Let us have groups G and P and a homomorphism f: G®Aut P. The mapping y of the group G onto the subset P^¢ of the group P by the rule y(g) = p is called a right quasi-homomorphism if for any g_i and g_j from G

y(gigj) = y(gi) [y(gj)] = pi (pj) = pk,

where p_i, p_j, p_kÎP^¢ and = f(g_i) [3]. The mapping f is called the accompanying homomorphism of right conjugation. Under the same initial conditions the mapping m is called a left quasi-homomorphism, accompanied by the homomorphism fof left conjugation, if

m(gigj) = [m(gi)] m(gj) = (pi) pj = pk.

At the right quasi-homomorphism y, in general, the image of G y(G) = P^¢Ì P is not a group, but P^¢ always contains the unit of the group P. The kernel H of the right quasi-homomorphism y of the group G into the group P is a subgroup in G; the index of this subgroup coincides with the power of y(G).

The mapping  of the group G onto the subset X of the set of all the right cosets of group P by its subgroup Q (Q < P) is called a generalized right quasi-homomorphism if for any g_i and g_j from G conditions (g_i) = Qp_i and (g_j) = Qp_j imply

(gigj) = Qpi · (Qpj) = Qpk,

where  Î Aut P, f: G ®Aut P is an accompanying homomorphism and Qp_i, Qp_j, Qp_k Î X.

The necessary and sufficient condition for the mapping of the group G onto the subset X of the set of all the right cosets of group P by its subgroup Q by the rule (g) = Qp to be a generalized right quasi-homomorphism is (Q) = p^-1Qp for any g ÎG and Qp = (g) [3].
 
 

Any group G^(P) of complete P-symmetry (P^¢ = P) can be derived from its generating group G and permutation group P by the following steps: 

        1) to find in G and in P all invariant subgroups H and Q for which there is the isomorphism of factor-groups S/H and P/Q (l: G/H ® P/Q) by the rule l(gH) = pQ; 

        2) to combine pair-wise each g^¢ of gH with each p^¢ of pQ = l(gH); 

        3) to introduce into the set of all these pairs the operation (1).

If Q = e (G^(P) is a junior group), then the isomorphism l: G/H ® P is, in fact, a homomorphism of the group G onto P (i.e. it is a representation of the group G) with the kernel H [1, 2].
 

Example 5. In the case when the generating group G = {a, b} (4) (we use Zamorzaev's symbols [2]) and the permutation group P = (e, p₁ = (1234), p₂ = (13)(24), p₃ = (1432)) @ 4, we have only three different junior groups of 4-symmetry. Their geometrical interpretations are represented by colored periodical mosaics (Figures 9 - 11).
 
 

Any group of `P-symmetry (with a finite group P) can be derived from its generating group G, knowing the kernel H of accompanying homomorphism f: G ® Aut P, by the following steps: 

        1) to find in P all subgroups Q and subsets P^¢, which are decomposed in right cosets by its subgroup Q, and in G all proper subgroups H^¢ (H^¢< G) with the index equal to the power of set of all the right cosets of P^¢by Q and for which there is the isomorphism l of factor-groups H/H¢¢ and P¢¢/Q (l: H/H¢¢ ® P¢¢/Q) by the rule l(gH¢¢) = pQ) where e  Q  P¢¢ÍP^¢ÍP, P¢¢ < P and H¢¢ = H^¢Ç H  H;

        2) to construct a generalized right quasi-homomorphism  of the group G onto the set of all the right cosets of P^¢ by the subgroup Q by the rule (gH^¢) = Qp and with accompanying homomorphism f: G ® Aut P with the kernel H; 

        3) to combine pair-wise each g^¢ of gH^¢ with each p^¢ of Qp = (g^¢); 

        4) to introduce into the set of all these pairs the operation (2).

If Q = e, then the mapping  is an ordinary right quasi-homomorphism and the universal method of deriving the groups of `P-symmetry becomes more simple and takes the form of the method of deriving the junior, semi-junior or pseudo-junior groups in dependence on P^¢ [3].
 
 

The natural left quasi-homomorphism m of the group G into the group W = `Õ_{gi Î G}P^g_i under which the automorphism operates on the elements w of m(G) by means of the left g-translations of their components is called an exact natural left quasi-homomorphism.

The necessary and sufficient condition for the mapping m of the group G onto the subset W¢ of the group W = `Õ_{gi Î G}P^g_i by the rule m(g) = w to be an exact natural left quasi-homomorphism is that (w_i)w_j = w_i^g_jw_j = w_k Î W^¢ for any g_jÎ G and w_j = m(g_j). Moreover, the necessary and sufficient condition for the exact natural left quasi-homomorphism m of the group G onto the subgroup W^¢ of W to be an ordinary homomorphism is W^¢ £ Diag W. We note that in the case of the exact natural left quasi-homomorphism m of the group G onto the subgroup W^¢ of the group W with the kernel Ker m = H the followings conditions are not compatible: 

        1) G/H @ W^¢),

        2) W^¢  Diag W [14].

Let us have the group G, the finite group W = Õ_{gi Î G}P^g_i, its subgroup V (V < W) and the exact isomorphic injection f of the group G into the subgroup of the group Aut W by the rule f(g) = . The mapping 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 quasi-homomorphism if for any g_i and g_j from G conditions (g_i) = w_iV and (g_j) = w_jV imply

(gigj) = (wiV)gj*wjV = wkV,

where w_iV, w_jV, w_kVÎX. Note that in the case of V = w₀ the mapping  is an ordinary exact natural left quasi-homomorphism.

The necessary and sufficient condition for the mapping of the group G onto the subset X of the set of all left cosets of the finite group W = Õ_{gi Î G}P^g_i by its subgroup V by the rule  (g) = wV to be a generalized exact natural left quasi-homomorphism is V^g = wVw^-1 for any g ÎG and wV = (g).

If V  W, then the generalized exact natural left quasi-homomorphism  of the group G onto the subset X of the set of all left cosets of group W by its subgroup V is an ordinary left quasi-homomorphism m of the group G onto the subgroup X of the factor-group W/V. In this case the mapping  is accompanied by homomorphism : G ® Aut W/V with the kernel H, where H @ F₀  < Aut W and F₀ is the set of the automorphisms of group W (and  Î Aut W) that generates the identical automorphism of factor-group W/V [36].

Any group G^(W_p) of W_p-symmetry with the finite group W can be derived from its finite generating group G and a group W = Õ_{gi Î G}P^g_i of multi-component permutations 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 l of factor-groups G₁/H and W₁/V₁ (l: G₁/H ® W₁/V by the rule l(Hg) = wV), where G₁£ G, W₁ £ Diag W and V₁ = V Ç Diag W £ W₁; 

        2) to construct a generalized exact natural left quasi-homomorphism  of the group G onto the set of all left cosets of W^¢ by the subgroup V by the rule (Hg) = wV and which preserves the correspondence between the elements of factor-groups G₁/H and W₁/V₁ obtained as the result of isomorphism l; 

        3) to combine pair-wise each g^¢ of Hg with each w^¢ of wV = (g^¢); 

        4) to introduce into the set of all these pairs the operation (3) [16-21, 28].

If V = w₀, where w₀ is the unit of the group W, then the mapping is an ordinary exact natural left quasi-homomorphism. In this case, the universal method of deriving the groups of W_p-symmetry becomes more simple and takes the form of method for deriving the semi-junior or pseudo-junior groups in dependence on W^¢, where W^¢ Ì W [14].

For the groups G^(W_p) of W_p-symmetry the polynomial symbol (the symbol is formed from more terms) was proposed in the form

        1) G is the generating group for G^(W_p);

        2) P is the initial group of permutations;

        3) W^¢ = {w| g^(w)Î G^(W_p)} Í W;

        4) V is the subgroup of W-identical transformations;

        5) H is the classical symmetry subgroup;

        6) G₁/H^¢/H is the trinomial symbol (the three-term symbol) for the group of P-symmetry G₁^(W₁) in accordance with [34, 2];

        7) W₁ / V₁ @ G₁/H, where V₁ = V Ç Diag W £ W₁ and W₁ = W^¢Ç Diag W.

In the case of semi-junior and pseudo-junior groups the polynomial symbol becomes simpler and takes the form

Example 6. The colored square represented in Figure 4 is described by the semi-junior group of W_p-symmetry G^(W_p) = (1w₀, 4w₁, 2w₂, 4^-1w₃), where w₀ = < e¹, e⁴, e², e^4-1> is unit of the group W = P¹ × P⁴ × P² × P^4-1 with P = (e, p = (12)), w₁ = < p¹, e⁴, p², e^4-1>, w₂ = < p¹, p⁴, p², p^4-1> and w₃ = < e¹, p⁴, e², p^4-1> . The polynomial abbreviated symbol of this group has the form 4/(2/1/1).

We note that the colored square represented in Figure 5 is described by the semi-junior group of W_p-symmetry G₁^(W_p) = (1w₀, 4w, 2w₀, 4^-1w), where w = < p¹, p⁴, p², p^4-1> ; the polynomial abbreviated symbol of this group has the form 4/(4/2/2). Moreover, the same colored square is described by the junior group of P-symmetry G^(P) = (1e, 4p, 2e, 4^-1p), with the three-term symbol 4/2/2. The group G₁^(W_p) may be formally considered as a group of P-symmetry.

The square with classical symmetry group G = C₄ may be painted with two colors by one more mode [35] (Figure 12).

Let us have groups G, P and W = `Õ_{gi Î G}P^g_i (where P^g_i @ P), the isomorphic injection f: G ® Aut W by the rule f(g) = , where the automorphism  makes the left g-translation of the components in w Î W (i.e. : w®w^g), and also the homomorphism t: G ® F £ Aut W (where t(g) =  and (w) = gwg^-1). The mapping a of the group G onto the subset W^¢ of the group W by the rule a(g) = w is called a crossed quasi-homomorphism, accompanied by the exact left translation of components and by the homomorphism t of right conjugation, if for any g_i and g_j from G

a(gigj) = [a(gi)]gj* [a(gj)] = wigj (wj) = wk,

We note that in the case of  = i (where i is the identical automorphism of group W for any g from G) the crossed quasi-homomorphism a is an ordinary exact natural left quasi-homomorphism; if w^g = w for all g Î G and w Î a(G), then a is right quasi-homomorphism accompanied by the homomorphism t of right conjugation.

In general, the image of G,  a(G) = W^¢Ì W is not a group, but W^¢ always contains the unit of the group W. The kernel H of crossed quasi-homomorphism a of the group G in the group W is a subgroup of the group G.

Let us have crossed quasi-homomorphism a with Ker a = H of the group G onto the subset W^¢ of the group W accompanied by exact left translations of components and by the homomorphism t of right conjugation with Ker t = H₁. Then: 

        1) the necessary and sufficient condition for the mapping a to send each left coset gH onto the only element w is that w^h = w for any h Î H = Ker a and w ÎW^¢; 

        2) if Ker a £ Ker t, then a sends each right coset Hg of group G by its subgroup H = Ker a onto the only element wÎ W^¢.

Moreover, if the kernel Ker a = H of the mapping a is an invariant of the group G and H £ Ker t, then: 

        a) the crossed quasi-homomorphism a sends each left coset gH onto the only element w Î W^¢; 
        b) the equalities w^h = w are valid for any h ÎH and for any w from W^¢; 

        c) by the accompanying homomorphism t: G ® Aut W all elements gh ÎgH generate the same automorphism .

Any semi-junior group G^(W_q) of W_q-symmetry can be derived from its generating group G and group W = `Õ_{gi Î G}P^g_i, knowing the kernel H₁ of accompanying the homomorphism t: G ® Aut W of right conjugation, by the following steps: 

        1) to construct a crossed quasi-homomorphism a of the group G onto the non-trivial subgroup W^¢ of W by the rule a(g) = w; 

        2) to combine pair-wise each g of G with each w = a(g); 

        3) to introduce into the set of all these pairs the operation (4) [27-29].
 
 

Example 7. The "ïndexed" figure represented in Figure 7 is described by the pseudo-junior group of W_q-symmetry G^(W_q) = (w₀1, w₀2, w4, w4^-1), where w₀ = < e¹, e⁴, e², e^4-1> is unit of the group W = P¹ × P⁴ × P² × P^4-1 with P = (e,  p₁ = (1234),  p₂ = (13)(24), p₃ = (1432)) @C₄ and w = < p₁¹, p₃⁴, p₁², p₃^4-1> . In this case (w) = w for any g_i from G = C₄.

We note that the "ïndexed" figure represented in Figure 8 is described by the semi-junior group of W_q-symmetry

Let us have the group G, the finite group W = Õ_{gi Î G}P^g_i (where P^g_i @ P), its subgroup V, the isomorphic injection f: G® Aut W by the rule f(g) =  (where : w ® w^g) and also the homomorphism t: G® Aut W (where t(g) =  and (w) = gwg^-1). The mapping  of the group G onto the subset X of the set of all right cosets of group W by its subgroup V is called a generalized crossed quasi-homomorphism, accompanied by exact left translation of components and by homomorphism t of right conjugation, if for any g_i and g_j from G from the conditions (g_i) = Vw_i and (g_j) = Vw_j it follows that

(gigj) = (Vwi)gj* (Vwj) = Vwk,

where Vw_i, Vw_j, Vw_kÎX and  = t(g_i) Î F £ Aut W.

We note that in the case of V = w₀ the generalized crossed quasi-homomorphism  is an ordinary crossed quasi-homomorphism. If t_g = i for any g from G, then the mapping  is a generalized exact natural left quasi-homomorphism. Moreover, if w^g = w for any g from G and w from W^¢= (G), then is a generalized right quasi-homomorphism accompanied by the homomorphism t of right conjugation.

Any middle group G^(W_q) of W_q-symmetry with the finite group W and the subgroup V of W-identical transformations can be derived from its finite generating group G and the group W = Õ_{gi ÎG}P^g_i of multi-component permutations by the following steps: 

        1) to find in W all proper -invariant subgroups V (w₀< V< W); 

        2) to construct a generalized crossed quasi-homomorphism  of the group G onto the set of all right cosets of the group W by the subgroup V by the rule (g) = Vw; 

        3) to combine pair-wise each g of G with each w^¢ of Vw = (g); 

        4) to introduce into the set of all these pairs the operation (4) [28-33].

Remark that in operation (4) there are two different automorphisms and , which are independent and, therefore, by consecutive action on the same w they commute.
 
 

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