Smashed and Twisted Wreath Products of Metagroups

In this article, nonassociative metagroups are studied. Different types of smashed products and smashed twisted wreath products are scrutinized. Extensions of central metagroups are studied.


Introduction
Nonassociative algebras compose a great area of algebra.In nonassociative algebra, noncommutative geometry, and quantum field theory, there frequently appear binary systems which are nonassociative generalizations of groups and related with loops, quasi-groups, Moufang loops, etc., (see References [1][2][3][4] and references therein).It was investigated and proved in the 20th century that a nontrivial geometry exists if and only if there exists a corresponding loop [1,5,6].
A multiplicative law of their canonical bases is nonassociative and leads to a more general notion of a metagroup instead of a group [11,20,21].The preposition "meta" is used to emphasize that such an algebraic object has properties milder than a group.By their axiomatics, metagroups satisfy the conditions of Equations ( 1)-(3) and rather mild relations (Equation ( 9)).They were used in References [20,21] for investigations of automorphisms, derivations, and cohomologies of nonassociative algebras.In the associative case, twisted and wreath products of groups are used for investigations not only in algebra but also in algebraic geometry, geometry, coding theory, and PDEs and their applications [22][23][24][25].Twisted structures also naturally appear in investigations in the G-N theory of wave propagation of the components of displacement, stress, temperature distribution, and change in the volume fraction field in an isotropic homogeneous thermoelastic solid with voids subjected to thermal loading due to laser pulse [26].
In this article, nonassociative metagroups are studied.Necessary preliminary results on metagroups are described in Section 2. Quotient groups of metagroups are investigated in Theorem 1. Identities in metagroups established in Lemmas 1, 2, and 4 are applied in Sections 3 and 4.
Different types of smashed products of metagroups are investigated in Theorems 3 and 4.Besides them, direct products are also considered in Theorem 2. They provide large families of metagroups (see Remark 2).
In Section 4, smashed twisted wreath products of metagroups and particularly also of groups are scrutinized.It appears that, generally, they provide loops (see Theorem 5).If additional conditions are imposed, they give metagroups (see Theorem 6).Their metaisomorphisms are investigated in Theorem 7. In Theorem 8 and Corollary 2, smashed splitting extensions of nontrivial central metagroups are studied.
All main results of this paper are obtained for the first time.They can be used for further studies of binary systems, nonassociative algebra cohomologies, structure of nonassociative algebras, operator theory and spectral theory over Cayley-Dickson algebras, PDEs, noncommutative analysis, noncommutative geometry, mathematical physics, and their applications in the sciences (see also the conclusions).

Nonassociative Metagroups
To avoid misunderstandings, we give necessary definitions.
A reader familiar with References [1,20,21] may skip Definition 1.For short, it will be written as a metagroup instead of a nonassociative metagroup.Definition 1.Let G be a set with a single-valued binary operation (multiplication) G 2 (a, b) → ab ∈ G defined on G satisfying the following conditions: There exists a neutral (i.e., unit) element e G = e ∈ G: eg = ge = g for each g ∈ G. ( If the set G with the single-valued multiplication satisfies the conditions of Equations ( 1) and ( 2), then it is called a quasi-group.If the quasi-group G satisfies also the condition of Equation (3), then it is called an algebraic loop (or in short, a loop).
The set of all elements h ∈ G commuting and associating with G is as follows: We call G a metagroup if a set G possesses a single-valued binary operation and satisfies the conditions of Equations ( 1)-( 3) and (ab)c = t(a, b, c)a(bc) for each a, b, and c in G, where t(a, b, c If G is a quasi-group satisfying the condition of Equation ( 9), then it will be called a strict quasi-group.
Then, the metagroup G will be called a central metagroup, if it satisfies also the following condition: for each a and b in G, where t 2 (a, b) ∈ C(G).
If H is a submetagroup (or a subloop) of the metagroup G (or the loop G) and gH = Hg for each g ∈ G, then H will be called almost normal.If, in addition, (gH)k = g(Hk) and k(gH) = (kg)H for each g and k in G, then H will be called a normal submetagroup (or a normal subloop respectively).
Henceforward, notations Inv l (a) = Div l (a, e) and Inv r (a) = Div r (a, e) will be used.
Elements of a metagroup G will be denoted by small letters; subsets of G will be denoted by capital letters.If A and B are subsets in G, then A − B means the difference of them: A − B = {a ∈ A : a / ∈ B}.Henceforward, maps and functions on metagroups are supposed to be single-valued unless otherwise specified.
Proof.The conditions of Equations ( 1) for each a and b in G. Using the condition of Equation (9) and the identities of Equations ( 14) and (15), we deduce the following: which leads to Equation (11).Let c = a\b; then, from the identities of Equations (11) and (14), it follows that (a\e)b = (e/a)t(e/a, a, a\e)(ac) = ((e/a)a)(a\b)t(e/a, a, a\e)/t(e/a, a, a\b) which provides Equations (12).Now, let d = b/a; then, the identities of Equations (11) and (15) imply that b(e/a) = (da)(a\e)/t(e/a, a, a\e) = (b/a)t(b/a, a, a\e)/t(e/a, a, a\e) which demonstrates Equation (13).
Lemma 2. Assume that G is a metagroup.Thenm for every a, a 1 , a 2 , and a 3 in G and p 1 , p 2 , and p 3 in C(G), we have the following: t(a, a\e, a)t(a\e, a, e/a) = e.
Proof.Since (a 1 a 2 )a 3 = t(a 1 , a 2 , a 3 )a 1 (a 2 a 3 ) and t(a 1 , a 2 , a 3 ) ∈ C(G) for every a 1 , a 2 , a 3 in G, then Therefore, for every a 1 , a 2 , a 3 in G and p 1 , p 2 , and p 3 in C(G), we infer the following: For each p ∈ C(G), a and b in G, because C(G) is the commutative group.Thus, t(p 1 a 1 , p 2 a 2 , p 3 a 3 ) = t(a 1 , a 2 , a 3 ).
Theorem 1.If G is a metagroup and C 0 is a subgroup in a center C(G) such that t(a, b, c) ∈ C 0 for each a, b, and c in G, then its quotient G/C 0 is a group.
Proof.As traditionally, the following notation is used: Inv r (A) = {x = e/a : a ∈ A} for subsets A and B in G. Then from the conditions of Equations (4)-( 8), it follows that, for each a, b, and c in G, the following identities take place: ((aC 0 )(bC 0 ))(cC 0 ) = (aC 0 )((bC 0 )(cC 0 )) and aC 0 = C 0 a.Evidently eC 0 = C 0 .In view of Lemmas 1 and 2 (aC 0 )\e = e/(aC 0 ), consequently, for each aC 0 ∈ G/C 0 a unique inverse (aC 0 ) −1 exists.Thus the quotient G/C 0 of G by C 0 is a group.Lemma 3. Let G be a metagroup, then Inv r (G) and Inv l (G) are metagroups.
Proof.At first, we consider Inv r (G).Let a 1 and a 2 belong to G.Then, there are unique e/a 1 and e/a 2 , since the map Inv r is single-valued (see Definition 1).Since Inv r • Inv l (a) = a and Inv l • Inv r (a) = a for each a ∈ G, then Inv r : G → G and Inv l : G → G are bijective and surjective maps.

Smashed Products and Smashed Twisted Products of Metagroups
Theorem 2. Let G j be a family of metagroups (see Definition 1), where j ∈ J, J is a set.Then, their direct product G = ∏ j∈J G j is a metagroup and Proof.It is given in Theorem 8 in Reference [21].
Remark 1.Let A and B be two metagroups, and let C be a commutative group such that where C m (A) denotes a minimal subgroup in C(A) containing t A (a, b, c) for every a, b, and c in A.
Using direct products, it is always possible to extend either A or B to get such a case.In particular, either A or B may be a group.On A × B, an equivalence relation Ξ is considered such that for every v in A, b in B, and γ in C.
Let φ : A → A(B) be a single-valued mapping, where A(B) denotes a family of all bijective surjective single-valued mappings of B onto B subjected to the conditions of Equations ( 31)- (34) given below.If a ∈ A and b ∈ B, then it will be written shortly b a instead of φ(a)b, where φ(a) : B → B. Let also be single-valued mappings written shortly as η, κ, and ξ correspondingly such that (cb and and for every u and v in A, b, and c in B, γ in C, where e denotes the neutral element in C and in A and B. We put for each of a 1 and a 2 in A and of b 1 and b 2 in B. The Cartesian product A × B supplied with such a binary operation of Equation (36) will be denoted by A φ,η,κ,ξ B.
Then, we put for each of a 1 and a 2 in A and of b 1 and b 2 in B. The Cartesian product A × B supplied with a binary operation of Equation (37) will be denoted by A φ,η,κ,ξ B. Theorem 3. Let the conditions of Remark 1 be fulfilled.Then, the Cartesian product A × B supplied with a binary operation of Equation (36) is a metagroup.Moreover, there are embeddings of A and B into Proof.The first part of this theorem was proven in Theorem 9 in Reference [21].(31).
and Equations (30) and (35), Theorem 4. Suppose that the conditions of Remark 1 are satisfied.Then, the Cartesian product A × B supplied with a binary operation of Equation (37) is a metagroup.Moreover, there exist embeddings of A and B into Proof.The conditions of Remark 1 imply that the binary operation of Equation (37) is single-valued.
We consider the following formulas: , where a 1 , a 2 , and a 3 are in A and where b 1 , b 2 , and b 3 are in B. Utilizing Equations (31)-( 35) and (37), we get the following: ) and Therefore Consequently, t(( Evidently, Equation (3) is a consequence of Equations (35) and (37).
Note that, if γ ∈ C, then Then, we seek a solution of the following equation: where From Equations (2) and (37), it follows that Definition 2. We call the metagroup A φ,η,κ,ξ B provided by Theorem 3 (or A φ,η,κ,ξ B by Theorem 4) a smashed product (or a smashed twisted product correspondingly) of metagroups A and B with smashing factors φ, η, κ, and ξ.Remark 2. From Theorems 2-4, it follows that, taking nontrivial η, κ, and ξ and starting even from groups with nontrivial C(G j ) or C(A), it is possible to construct new metagroups with nontrivial C(G) and ranges t G (G, G, G) of t G that may be infinite.
With suitable smashing factors φ, η, κ, and ξ and with nontrivial metagroups or groups A and B, it is easy to get examples of metagroups in which e/a = a\e for an infinite family of elements a in A φ,η,κ,ξ B or in A φ,η,κ,ξ B. Evidently, smashed products and smashed twisted products (see Definition 2) are nonassociative generalizations of semidirect products.Combining Theorems 3 and 4 with Lemmas 3 and 4 provides other types of smashed products by taking b1 2 b 1 on the right sides of Equations (36) and (37), correspondingly, etc.

Smashed Twisted Wreath Products of Metagroups
Lemma 5. Let D be a metagroup and A be a submetagroup in D.Then, there exists a subset V in D such that D is a disjoint union of vA, where v ∈ V, that is, Proof.The cases A = {e} and A = D are trivial.Let A = {e} and A = D, and let C(D) be a center of D. From the conditions of Equations ( 4)-( 8), it follows that z ∈ C(D) ∩ A implies z ∈ C(A).
According to Equation (2), s = a/b; hence, a ψ = a/a τ .From Equation (13) ,it follows that a/b = a(e/b)t(e/b, b, b \ e)/t(a/b, b, b\e); consequently, by Lemma 2, s = aa e/τ t(a e/τ , a τ , a τ\e )/t(aa e/τ , a τ , a τ\e ). (55) Notice that the metagroup need not be power-associative.Then, e/s and s\e can be calculated with the help of the identity of Equation (11).
From Equations (64) and (65), it follows that the metagroup D acts on V transitively by right shift operators R y , where R y a = ay for each a and y in D. Therefore, we put (a τ ) [c] := (a τ c) τ f or each a and c in D. (68) Then from Equations (64), ( 65), (68), and (9) and Lemma 2, we deduce that, for each a, c, and d in D In particular, (a τ ) [e] = a τ for each a ∈ D. Next, we put e τ = b * .It is convenient to choose b * = e.Hence, b [s] * = (e τ ) [s] = (e τ s) τ = s τ = e = e τ for each s ∈ A. Thus, the submetagroup A is the stabilizer of e and Equation (68) implies that e [s] = e and e [q] = q f or each s ∈ A and q ∈ V.
(70) Remark 4. Let B and D be metagroups, A be a submetagroup in D, and V be a right transversal of A in D.
Let also the conditions of Equations ( 28)-( 35) be satisfied for A and B. By Theorem 2, there exists a metagroup It contains a submetagroup is a support of f ∈ F and card(Ω) denotes the cardinality of a set Ω.
Let T h f = f h for each f ∈ F and h : V → A. We put where J : (see also Equations (52) and (68)).Hence, since v e\e = v and s(e, v) = e.Lemma 6.Let the conditions of Remark 4 be satisfied.Then, for each of f and f 1 in F and of d and where w j = w j (d, Proof.Equations (72) and (33) imply the identity of Equation (75).
Then, we seek a solution x ∈ C (or x ∈ C * respectively) of the equation xa = b.It is equivalent to two equations: It remains to verify that F (or F * respectively) is an almost normal subloop in C (or C * respectively).Assume that Using the embedding C V → F and Equation (35), we infer that (d 1 , f 1 )F = F(d 1 , f 1 ), since F {d 1 } = F by Equation (68), Lemma 5, and Equation (30).It can be verified similarly that F * is the almost normal subloop in C * .Definition 5.The product Equation (86) in the loop C (or C * ) of Theorem 5 is called a smashed twisted wreath product of D and F (or a restricted smashed twisted wreath product of D and F * respectively) with smashing factors φ, η, κ, and ξ, and it will be denoted by C = D∆ φ,η,κ,ξ F (or C * = D∆ φ,η,κ,ξ F * respectively).The loop C (or C * ) is also called a smashed splitting extension of F (or of F * respectively) by D. Theorem 6.Let the conditions of Remark 4 be satisfied and C m (D) ⊆ C, where C is as in Equation (28) for each f ∈ F, v ∈ V, and d and d 1 in D, since C ⊆ C(A) by Equation (28).Next, we consider the following products: and Then, Equations (86), (90), and (33)- (35) imply that From Equations (88), (89), (76), and (35), we infer that where w 3 = w 3 (d 1 , d 2 , v).Therefore, from Equations (91) and (92), we infer that where for every f , Definition 6.Let P 1 and P 2 be two loops with centers C(P 1 ) and C(P 2 ).Let also for each a and b in P 1 , where ν(a, b) ∈ C(P 2 ).Then, µ will be called a metamorphism of P 1 into P 2 .If in addition µ is surjective and bijective, then it will be called a metaisomorphism and it will be said that P 1 is metaisomorphic to P 2 .
Then, P 1 is metaisomorphic to P 2 and P * 1 to P * 2 .
This implies that { f Therefore, we deduce using Equation (107) that [(e, f 0 ), (e/d 0 , e)] = (e, w f 0 ), where w = ξ((e, f 0 ), (e/d 0 , e))ξ((d 0 , e), (e/d 0 , f 0 )) ξ((e, e/ f 0 ), (e, ( f 0 ) 2 ))/t F * (e/ f 0 , f 0 , f 0 ), Proof.We take d 0 = d 1 in the proof of Theorem 8; thus, c 1 = (d 1 \e, e), c 2 = (d 2 \e, f 0 ), and c j = (d j \e, e) for each j ≥ 3. Therefore Equations (66), (108), and (35)  Example 1. Assume that A is a unital algebra over a commutative associative unital ring F supplied with a scalar involution a → ā so that its norm N and trace T maps have values in F and fulfil conditions: for each a and b in A. We remind that, if a scalar f ∈ F satisfies the condition ∀a ∈ A f a = 0 ⇒ a = 0, then such element f is called cancelable.For such a cancelable scalar f , the Cayley-Dickson doubling procedure induces a new algebra C(A, f ) over F such that for each a and b in A. Such an element l is called a doubling generator.From Equations (113)-( 115), it follows that ∀a ∈ A, ∀b ∈ A T(a) = T(a + bl) and N(a ) is a generalized Cayley-Dickson algebra, when F is not a field, or a Cayley-Dickson algebra, when F is a field.
If the doubling procedure starts from A = F1 =: A 0 , then A 1 = C(A, f 1 ) is a * -extension of F. If A 1 has a basis {1, u} over F with the multiplication table u 2 = u + w, where w ∈ F and 4w + 1 = 0, with the involution 1 = 1, ū = 1 − u, then A 2 is the generalized quaternion algebra and A 3 is the generalized octonion (Cayley-Dickson) algebra.
Particularly, for F = R and f n = 1 for each n by A r the real Cayley-Dickson algebra with generators i 0 , ..., i 2 r −1 will be denoted such that i 0 = 1, i 2 j = −1 for each j ≥ 1, and i j i k = −i k i j for each j = k ≥ 1.Note that the Cayley-Dickson algebra A r for each r ≥ 3 is nonassociative, for example, (i 1 i 2 )i 4 = −i 1 (i 2 i 4 ), etc.Moreover, for each r ≥ 4, the Cayley-Dickson algebra A r is nonalternative (see References [7,11,12]).Frequently, ā is also denoted by a * or ã.
Then, one can take a Cayley-Dickson algebra A n over a commutative associative unital ring R of characteristic different from two such that A 0 = R, n ≥ 2. There are basic generators i 0 , i 1 , ..., i 2 n −1 , where i 0 = 1.Choose Ψ as a multiplicative subgroup contained in the ring R such that f j ∈ Ψ for each j = 0, ..., n.Put G n = {i 0 , i 1 , ..., i 2 n −1 } × Ψ.Then, G n is a central metagroup because, in this case, Ψ is commutative.
Example 2.More generally, suppose that H is a group such that Ψ ⊂ H, with relations hi k = i k h and (hg)i k = h(gi k ) for each k = 0, 1, ..., 2 n − 1 and each h and g in H.Then, G n = {i 0 , i 1 , ..., i 2 n −1 } × H is also a metagroup.If the group H is noncommutative, then the latter metagroup can be noncentral (see the condition of Equation (10) in Definition 1).Utilizing the notation of Example 1, we get that the Cayley-Dickson algebra A ∞ over the real field R with f n = 1 for each n provides a pattern of a metagroup G ∞ = {i j , − i j : 0 ≤ j ∈ Z}, where Z denotes the ring of integers.
Example 3. Certainly, in general, metagroups need not be central.On the other hand, if a metagroup is associative, then it is a group [1].Apparently, each group is a metagroup also.For a group G, its associativity evidently means that t G (a, b, c) = e [1].
From the given metagroups, new metagroups can be constructed using their direct, semidirect products, smashed products, and smashed twisted wreath products.Therefore, there are abundant families of noncentral metagroups and also of central metagroups different from groups.

Conclusions
The results of this article can be used for further studies of metagroups, quasi-groups, loops, and noncommutative manifolds related with them.Besides applications of metagroups, loops, and quasi-groups outlined in the introduction, it is interesting to mention possible applications in mathematical coding theory and classification of information flows and their technological implementations [28][29][30] because, frequently, codes are based on binary systems.Moreover, twisted products are used for creating complicated codes [22].In view of this, to study creating more complicated codes with the help of smashed twisted products of metagroups, Equations (86) and (94)-(96) provide additional options in the nonassociative case in comparison with the associative case.
Wreath products of groups are used for studies of varieties [24], so it will be interesting to investigate noncommutative varieties using metagroups.Then, twisted products are utilized for investigations of Lie groups and semi-Riemann manifolds [23,25].Therefore, we will study their nonassociative metagroup analogs that can be used in noncommutative geometry and quantum field theory [16,[31][32][33][34][35] because Lie groups and manifolds are actively used in these areas.

For
each a and b in G, there is a unique x ∈ G with ax = b (1) and a unique y∈ G exists satisfying ya = b, which are denoted by x = a\b = Div l (a, b) and y = b/a = Div r (a, b) correspondingly,

Lemma 1 .
If G is a metagroup, then for each a and b ∈ G, the following identities are fulfilled: b\e = (e/b)t(e/b, b, b\e) e/b = (e/b)(b(b\e)) = (b\e)/t(e/b, b, b\e) Then, for each a, b, x, and y in G, the equations â • x = b and ŷ • â = b are equivalent to (e/x)(e/a) = e/b and (e/a)(e/y) = e/b, respectively.That is, x = (e/b)/(e/a) and ŷ = (e/a)\(e/b) are unique.On the other hand, e/e = e and ê • b = e/b = b • ê = b for each b ∈ G.

Definition 4 .
Suppose that the conditions of Remark 4 are satisfied and on the Cartesian product C = D × F (or C * = D × F * ) a binary operation is given by the following formula:

Theorem 5 .
for every d and d 1 in D, f and f 1 in F (or F * respectively), and v ∈ V. Let C, C * , D, F, and F * be the same as in Definition 4.Then, C and C * are loops and there are natural embeddings D → C, F → C, D → C * , and F * → C * such that F (or F * ) is an almost normal subloop in C (or C * respectively).Proof.The operation of Equation (86) is single-valued.Let a = (d, f ) and b = (d 0 , f 0 ), where d and d 0 are in D and where f and f 0 are in F (or F * ).The equation ay = b is equivalent to dd 2 = d 0 and

5 )
Then, C and C * supplied with the binary operation of Equation (86) are metagroups.Proof.In view of Theorem 5, C and C * are loops.To each element b in B, there corresponds an element {b(v) : ∀v ∈ V, b(v) = b} in F which can be denoted by b also.From the conditions of Equations (29)-(35), we deduce that γ a = γ and f γ = f for every γ ∈ C and a ∈ A. (87) Hence, Equations (87) and (86) imply that (C(A), C(F)) ⊆ C(C).On the other hand, w 1 = γ τ with γ ∈ C m (D) and w 2 = γ 3 /(γ with γ 1 ,...,γ 5 in C m (D) (see Equation (84)); hence, the condition C m (D) ⊂ C implies that Equation (76) simplifies to

d 1 ,Remark 5 .
d 2 in D, and v ∈ V. Then from Equation (93), C(F) = (C(B)) V (see Theorem 2) and Equation (28), it follows that the loops C and C * satisfy the condition of Equation (9), since (C, C V ) ⊆ C(C).Thus, C and C * are metagroups.Generally, if A = {e} and A = D, B, φ, η, κ, and ξ are nontrivial, where A, B, and D are metagroups or particularly may be groups, then the loops C and C * of Theorem 5 can be non-metagroups.If Equation (35) drops the conditions ξ((e, e), (v, b)) = e and ξ((v, b), (e, e)) = e for each v ∈ V and b ∈ B, then the proofs of Theorems 3-5 demonstrate that C 1 and C 2 are strict quasi-groups and that C and C * are quasi-groups.
[a, b] = (e/a)((e/b)(ab)) for each a and b in C * .Proof.Let d 0 be an arbitrary fixed element in D − C(D).Assume that A is a submetagroup in D such that A is generated by d 0 and a subgroup C 0 contained in a center C(D) of D, C m (D) ⊆ C 0 ⊆ C(D), where C m (D) is a minimal subgroup in a center C(D) of D such that t D (a, b, c) ∈ C m (D) for each of a, b, and c in D. Therefore, a k a n = p(k, n, a)a k+n (105) for each a ∈ A, k, and n in C = {0, −1, 1, −2, 2, ...}, where the following notation is used: a 2 = aa, a n+1 = a n a and a −n = e/a n , and a 0 = e for each n ∈ N and p(k, n, a) ∈ C m (A).Hence, in particular, A is a central metagroup.Then, d 0 C m (A) is a cyclic element in the quotient group A/C m (A) (see Theorem 1).Then, we choose a central metagroup B generated by an element b 0 and a commutative group C 1 such that b 0 / ∈ C 1 , C m (D) → C 1 and C(A) → C 1 and the quotient group B/C m (B) is of finite order l > 1.Then, let φ : A → A(B) satisfy the condition of Equation ), where C m (B) → C, C m (A) → C, C 0 → C, and C 1 → C. With these data, according to Theorem 6, C * is a metagroup, since C m (D) → C 1 and C m (D) → C 0 .That is, C * is a smashed splitting extension of the central metagroup F * by D.

{v} 0 :
v ∈ V}C(F * ) generates F * .Evidently, [v(d 0 \e)] τ = e for each v ∈ V − {e}, since d 0 \e ∈ A and the following conditions s ∈ D, sq ∈ A, and q ∈ A imply that s ∈ A because A is the submetagroup in D. Note that e/d = (d \ e)/t A (e/d, d, d\e) for each d ∈ A by Equation (11); consequently, s(d, e) = dt A (e/d, d, d\e).On the other hand, t A (a, b, c) ∈ C for each of a, b, and c in A and f γ 0 = f 0 f or each γ ∈ C (107) by Equation (87); hence, f {d 0 } 0 (e) = φ(d 0 )b 0 = b 2 0 , and consequently,

Corollary 2 .
and C j → C for each j ∈ {0, 1}.Therefore, Equations (107), (108), and (88) imply that cF * = F * c and c[F * , C * ]C(F * ) = [F * , C * ]C(F * )c for each c ∈ C * .Hence, [F * , C * ]C(F * ) ⊆ F * .Taking H = F * ,we get the assertion of this theorem.Let the conditions of Theorem 8 be satisfied and D be generated by C m (D) and at least two elements d 1 , d 2 ,... such that d 1 = e and [d 2 \e, d 1 \e] = e.Then, the smashed splitting extension C * can be generated by C(F * ) and elements c 1 , c 2 ,... such that d j \e ∈ F * c j for each j.