The recent generalizationsof colored symmetryLungu AlexandruState University of Moldova str. A. Mateevici 60 MD 2009 Chisinau Moldova lungu@usm.md
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 quasihomomorphisms and their generalizations. The principal properties of diverse quasihomomorphic 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; Wsymmetry
groups; action of automorphism; generalizations of homomorphism; quasihomomorphic
reflections; wreath products.
AMS Subject Classification: Primary: 20H15; 20E36; 20B25; 20A05; 51M20; 51N30; Secondary: 52; 82.
1. Introduction 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 Psymmetry of A.M.Zamorzaev [13]. In the case of Psymmetry, 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 WittkeGarrido [4] or complex [5] symmetry are not included in the scheme of A.M.Zamorzaev's Psymmetry. 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, 621, 28, 33, 35]. On the other hand, V.A.Koptsik in [22] proposed the notion of Qsymmetry. The essence of Qsymmetry consists actually in the following: the transformation of Qsymmetry 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 Qsymmetry (`Psymmetry [2325]), we shall get a generalization named W_{q}symmetry [2633]. The methods of deriving the groups of Psymmetry
of different types [1, 2] are based
on homomorphic mapping and its properties. The solution of analogous problems
for`P]
and Wsymmetry demands the generalization of homomorphisms as the
right quasihomomorphism [23, 3],
the natural left quasihomomorphism [14], and the
crossed quasihomomorphism [27]. Moreover, it requires
the investigation of some their properties.
2. The crossed standard Cartesian wreath product of groups 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_{gi Î G}P^{gi}, where P^{gi }@ P. Moreover, construct the isomorphic injection f: G®Aut W by the rule f(g) = (where the automorphism makes the left gtranslation of the components in w ÎW, i.e. : w®w^{g}), and also the homomorphism t: G ® F £ Aut W (where t(g) = and (w) = gwg^{1}). The set G^{*} of pairs wg (where w ÎW and gÎG) forms a group with the operation w_{i}g_{i}* w_{j}g_{j} = w_{k}g_{k}, where g_{k} = g_{i}g_{j} and w_{k} = w_{i}^{gj}(w_{j}). We call G^{*} the crossed standard Cartesian wreath product of groups P and G, accompanied with the homomorphism t : G ®Aut W by the rule t(g_{i}) = , 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.
3. The essence of the recent generalizations of color symmetry Ascribe to each point of a geometrical figure F with the discrete symmetry group G at least one index, which means a nongeometrical feature, from the set N = {1,2,...,m}, and fix a certain transitive group P of the permutations of these indexes. The transformation of Psymmetry 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 Psymmetry
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 Psymmetry 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,
p_{1}
= (1234), p_{2} = (13)(24), p_{3} = (1432)).
Then g^{(p)} = gp = 4p_{1}
is a transformation of Psymmetry of the colored square represented
in Figure 1.
Figure 1 The group of above Psymmetry 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 `Psymmetry 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^{(`P)} of transformations of `Psymmetry of any "indexed'' geometrical figure F^{(N)} forms a group with the operation
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.
Figure 2 In this case, the "ïndexed" figures represented in
Figure 3a) and Figure 3b) are described from the viewpoint of Psymmetry
by the same group with the group/subgroup symbol 4/1.
Figure 3 From the viewpoint of `Psymmetry 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^{gi},
where P^{gi
}@ P.
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 < p^{1}, e^{4}, p^{2}, e^{41}> is a transformation of W_{p}symmetry of the colored square represented in Figure 4.
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 < p^{1}, p^{4}, p^{2}, p^{41}> exists too, which may be formally considered as a transformation of Psymmetry (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 Psymmetry g^{(p)} = gp = 4p describe the same colored square (Figure 5).
Figure 5 The groups G^{(Wp)} 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:
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}(M_{1}) of the figure F^{(N)} (where M_{1} 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^{gk} ("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^{(Wq)} of transformations of W_{q}symmetry of the given "indexed'' figure F^{(N)} forms a group with the operation
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;
Figure 6 let's have the groups G = C_{4} = 4 = (1, 4, 2, 4^{1}) and P = (e, p_{1} = (1234), p_{2} = (13)(24), p_{3} = (1432)) @C_{4}. Then g^{(w)} = wg = < p_{1}^{1}, p_{3}^{4}, p_{1}^{2}, p_{3}^{41}> 4 is a transformation of W_{q}symmetry of the "ïndexed" figure represented in Figure 7.
Figure 7 We note that in the case when G = C_{4v}
= 4 ·m = (1, 4, 2, 4^{1}, m_{1},
m_{2},
m_{3},
m_{4})
and P@C_{4}, the transformation
of
W_{q}symmetry
g^{(w)} = wg
= <
p_{2}^{1},
p_{2}^{4},
p_{2}^{2},
p_{2}^{41},
p_{2}^{m1},
p_{2}^{m2}, p_{2}^{m3},
p_{2}^{m4} > 4
exists too, which may be considered formally as a transformation of `Psymmetry.
This transformation and the transformation of `Psymmetry
g^{(p)}
= pg = p_{2}4 describe the same "ïndexed" figure
F^{(N)}
represented in Figure 8.
Figure 8 The groups G^{(Wq)} 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}) =:
4. On the classification and the general structure of Wsymmetry groups 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 ®gw_{0}, where w_{0} is the unit of the group W, and w 1w, where 1 is the unit of the group G (g w_{0}g 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 Widentical transformations of the group G^{(W)}. The group G^{(W)} is called senior, junior or Vmiddle if w_{0} < V = W^{¢} = W, w_{0} = V < W^{¢} = W or w_{0} < V < W^{¢} = W, respectively. If W^{¢} is a nontrivial subgroup of W, then the group G^{(W)} is called W^{¢}semisenior, W^{¢}semijunior or (W^{¢}, V)semimiddle according to the cases when w_{0} < V = W^{¢}, w_{0} = V < W^{¢} or w_{0} < V < W^{¢}. If W^{¢}ÌW, but W^{¢} is not a group, the group G^{(W)} is called W^{¢}pseudojunior or (W^{¢}, V)pseudomiddle when w_{0} = V Ì W^{¢} or w_{0} < V Ì W^{¢}. Let G^{(Wp)} be a group of W_{p}symmetry with the initial group P, generating group G, subset W^{¢} = {w  g^{(w)} ÎG^{(Wp)}}, symmetry subgroup H and the subgroup V of Widentical transformations. Then: 1) the mapping f of the group G^{(Wp)} onto the group G by the rule f[g^{(w)}] = g is homomorphic with the kernel V; 2) the group G^{(Wp)} contains as its subgroup the group G_{1}^{(W1)} of Psymmetry (which is determined by initial group P of permutations, where W_{1}£ Diag W @P and W_{1}ÌW^{¢}) from the family with the generating group G_{1} (G_{1 }£ G), with the symmetry subgroup H (where H G_{1} but H ¹ G) and with the subgroup V_{1} of Widentical transformations (where V_{1} = 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^{(Wp)}; 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_{i}V and w_{j}V [1820, 28]. Let G^{(Wq)} be a group of W_{q}symmetry with the generating group G, permutation group P, (i.e. W = `Õ_{gi Î G}P^{gi}, where P^{gi }@ P), subset W^{¢} = {w  g^{(w) }Î G^{(Wq)}}, with the kernel H_{1} of accompanying homomorphism t: G ® Aut W, symmetry subgroup H and the subgroup V of Widentical transformations. In this case the following conditions are satisfied: 1) the mapping f of the group G^{(Wq)} onto the generating group G by the rule f[g^{(w)}] = g is homomorphic with the kernel V; 2) the group G^{(Wq)} contains as subgroup the group H_{1}^{(Wp)} of W_{p}symmetry (which is determined by the initial group P of permutations) from the family with the generating group H_{1}, with the symmetry subgroup H^{¢} (where H^{¢} = H Ç H_{1}) and with the same subgroup V of Widentical transformations; 3) the group G^{(Wq)} contains as a subgroup the group G_{1}^{(W1)} of `Psymmetry (which is determined by initial group P of permutations) from the family with the generating group G_{1} (where G_{1} £ G), with the same kernel H_{1} of the accompanying homomorphism and with the set W_{1} = {w  g^{(w) }Î G_{1}^{(W1)}} of multicomponent permutations, where W_{1} = W^{¢ }Ç Diag W. If W is a finite group of multicomponent permutations, then: 1) V^{g} = V for any transformation g^{(w)} from G^{(Wq)}, i.e. V is invariant subgroup; 2) (V)
= (w_{i}^{(gj)})^{1}Vw_{i}^{gj}
for transformations g_{i}^{(wi)}
and g_{j}^{(wj)} from G^{(Wq)}
[2628].
5. On the right and left quasihomomorphisms 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 quasihomomorphism if for any g_{i} and g_{j} from G
At the right quasihomomorphism 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 quasihomomorphism 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 quasihomomorphism if for any g_{i} and g_{j} from G conditions (g_{i}) = Qp_{i} and (g_{j}) = Qp_{j} imply
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 quasihomomorphism is
(Q)
= p^{1}Qp for any g ÎG
and Qp = (g)
[3].
6. On the methods for deriving groups of P and `Psymmetry Any group G^{(P)} of complete Psymmetry (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 factorgroups S/H and P/Q (l: G/H ® P/Q) by the rule l(gH) = pQ; 2) to combine pairwise 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, p_{1} =
(1234), p_{2} = (13)(24), p_{3} = (1432))
@
4, we have only three different junior groups of 4symmetry. Their geometrical
interpretations are represented by colored periodical mosaics (Figures
9  11).
Figure 9: Geometrical interpretation of the junior
group {a, b} (4^{(4)}).
Figure 10: Geometrical interpretation of the junior
group {a, b} (4^{(4)}).
Figure 11: Geometrical interpretation of the junior group {a^{(2)}, b^{(2)}} (4^{(4)}).
Any group of `Psymmetry (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 factorgroups 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 quasihomomorphism 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 pairwise 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 quasihomomorphism and the universal method of deriving
the groups of `Psymmetry becomes more
simple and takes the form of the method of deriving the junior, semijunior
or pseudojunior groups in dependence on P^{¢}
[3].
7. On the generalized exact natural left quasihomomorphisms and the groups of W_{p}symmetry The natural left quasihomomorphism m of the group G into the group W = `Õ_{gi Î G}P^{gi} under which the automorphism operates on the elements w of m(G) by means of the left gtranslations of their components is called an exact natural left quasihomomorphism. 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^{gi} by the rule m(g) = w to be an exact natural left quasihomomorphism is that (w_{i})w_{j} = w_{i}^{gj}w_{j} = w_{k} Î W^{¢} for any g_{j}Î G and w_{j} = m(g_{j}). Moreover, the necessary and sufficient condition for the exact natural left quasihomomorphism 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 quasihomomorphism 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^{gi}, 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 quasihomomorphism if for any g_{i} and g_{j} from G conditions (g_{i}) = w_{i}V and (g_{j}) = w_{j}V imply
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^{gi} by its subgroup V by the rule (g) = wV to be a generalized exact natural left quasihomomorphism is V^{g} = wVw^{1} for any g ÎG and wV = (g). If V W, then the generalized exact natural left quasihomomorphism 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 quasihomomorphism m of the group G onto the subgroup X of the factorgroup W/V. In this case the mapping is accompanied by homomorphism : G ® Aut W/V with the kernel H, where H @ F_{0} < Aut W and F_{0} is the set of the automorphisms of group W (and Î Aut W) that generates the identical automorphism of factorgroup W/V [36]. Any group G^{(Wp)} 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^{gi} of multicomponent 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 factorgroups G_{1}/H and W_{1}/V_{1} (l: G_{1}/H ® W_{1}/V by the rule l(Hg) = wV), where G_{1}£ G, W_{1 }£ Diag W and V_{1} = V Ç Diag W £ W_{1}; 2) to construct a generalized exact natural left quasihomomorphism 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 factorgroups G_{1}/H and W_{1}/V_{1} obtained as the result of isomorphism l; 3) to combine pairwise each g^{¢} of Hg with each w^{¢} of wV = (g^{¢}); 4) to introduce into the set of all these pairs the operation (3) [1621, 28]. If V = w_{0}, where w_{0} is the unit of the group W, then the mapping is an ordinary exact natural left quasihomomorphism. 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 semijunior or pseudojunior groups in dependence on W^{¢}, where W^{¢} Ì W [14]. For the groups G^{(Wp)} 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^{(Wp)}; 2) P is the initial group of permutations; 3) W^{¢} = {w g^{(w)}Î G^{(Wp)}} Í W; 4) V is the subgroup of Widentical transformations; 5) H is the classical symmetry subgroup; 6) G_{1}/H^{¢}/H is the trinomial symbol (the threeterm symbol) for the group of Psymmetry G_{1}^{(W1)} in accordance with [34, 2]; 7) W_{1} / V_{1} @ G_{1}/H, where V_{1} = V Ç Diag W £ W_{1} and W_{1} = W^{¢}Ç Diag W. In the case of semijunior and pseudojunior groups the polynomial symbol becomes simpler and takes the form
Example 6. The colored square represented in Figure 4 is described by the semijunior group of W_{p}symmetry G^{(Wp)} = (1w_{0}, 4w_{1}, 2w_{2}, 4^{1}w_{3}), where w_{0} = < e^{1}, e^{4}, e^{2}, e^{41}> is unit of the group W = P^{1} × P^{4} × P^{2} × P^{41} with P = (e, p = (12)), w_{1} = < p^{1}, e^{4}, p^{2}, e^{41}>, w_{2} = < p^{1}, p^{4}, p^{2}, p^{41}> and w_{3} = < e^{1}, p^{4}, e^{2}, p^{41}> . 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 semijunior group of W_{p}symmetry G_{1}^{(Wp)} = (1w_{0}, 4w, 2w_{0}, 4^{1}w), where w = < p^{1}, p^{4}, p^{2}, p^{41}> ; 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 Psymmetry G^{(P)} = (1e, 4p, 2e, 4^{1}p), with the threeterm symbol 4/2/2. The group G_{1}^{(Wp)} may be formally considered as a group of Psymmetry. The square with classical symmetry group G = C_{4} may be painted with two colors by one more mode [35] (Figure 12).
Figure 12: Geometrical interpretation of the pseudojunior
group
8. The crossed quasihomomorphisms and the semijunior groups of W_{q}symmetry Let us have groups G, P and W = `Õ_{gi Î G}P^{gi} (where P^{gi }@ P), the isomorphic injection f: G ® Aut W by the rule f(g) = , where the automorphism makes the left gtranslation 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 quasihomomorphism, 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
We note that in the case of = i (where i is the identical automorphism of group W for any g from G) the crossed quasihomomorphism a is an ordinary exact natural left quasihomomorphism; if w^{g} = w for all g Î G and w Î a(G), then a is right quasihomomorphism 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 quasihomomorphism a of the group G in the group W is a subgroup of the group G. Let us have crossed quasihomomorphism 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 = H_{1}. 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
quasihomomorphism
a sends each left coset gH
onto the only element w Î W^{¢};
c) by the accompanying homomorphism t: G ® Aut W all elements gh ÎgH generate the same automorphism . Any semijunior group G^{(Wq)} of W_{q}symmetry can be derived from its generating group G and group W = `Õ_{gi Î G}P^{gi}, knowing the kernel H_{1} of accompanying the homomorphism t: G ® Aut W of right conjugation, by the following steps: 1) to construct a crossed quasihomomorphism a of the group G onto the nontrivial subgroup W^{¢} of W by the rule a(g) = w; 2) to combine pairwise each g of G with each w = a(g); 3) to introduce
into the set of all these pairs the operation (4) [2729].
Example 7. The "ïndexed" figure represented in Figure 7 is described by the pseudojunior group of W_{q}symmetry G^{(Wq)} = (w_{0}1, w_{0}2, w4, w4^{1}), where w_{0} = < e^{1}, e^{4}, e^{2}, e^{41}> is unit of the group W = P^{1} × P^{4} × P^{2} × P^{41} with P = (e, p_{1} = (1234), p_{2} = (13)(24), p_{3} = (1432)) @C_{4} and w = < p_{1}^{1}, p_{3}^{4}, p_{1}^{2}, p_{3}^{41}> . In this case (w) = w for any g_{i} from G = C_{4}. We note that the "ïndexed" figure represented in Figure 8 is described by the semijunior group of W_{q}symmetry
9. The generalized crossed quasihomomorphisms and the middle groups of W_{q}symmetry Let us have the group G, the finite group W = Õ_{gi Î G}P^{gi} (where P^{gi }@ 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 quasihomomorphism, 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
We note that in the case of V = w_{0} the generalized crossed quasihomomorphism is an ordinary crossed quasihomomorphism. If t_{g} = i for any g from G, then the mapping is a generalized exact natural left quasihomomorphism. Moreover, if w^{g} = w for any g from G and w from W^{¢}= (G), then is a generalized right quasihomomorphism accompanied by the homomorphism t of right conjugation. Any middle group G^{(Wq)} of W_{q}symmetry with the finite group W and the subgroup V of Widentical transformations can be derived from its finite generating group G and the group W = Õ_{gi ÎG}P^{gi} of multicomponent permutations by the following steps: 1) to find in W all proper invariant subgroups V (w_{0}< V< W); 2) to construct a generalized crossed quasihomomorphism 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 pairwise each g of G with each w^{¢} of Vw = (g); 4) to introduce into the set of all these pairs the operation (4) [2833]. 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.
References [1] Zamorzaev A.M., On the groups of quasisymmetry (Psymmetry). Kristallografiya, 1967, 12, no.5, p.819825 (in Russan).

