On Smarandache Bryant Schneider Group of A Smarandache Loop

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1 International J.Math. Combin. Vol.2 (2008), 5-63 On Smarandache Bryant Schneider Group o A Smarandache Loop Tèmító. pé. Gbó. láhàn Jaíyéọlá (Department o Mathematics o Obaemi Awolowo University, Ile Ie, Nieria.) tjayeola@oauie.edu.n Abstract: The concept o Smarandache Bryant Schneider Group o a Smarandache loop is introduced. Relationship(s) between the Bryant Schneider Group and the Smarandache Bryant Schneider Group o an S-loop are discovered and the later is ound to be useul in indin Smarandache isotopy-isomorphy condition(s) in S-loops just like the ormal is useul in indin isotopy-isomorphy condition(s) in loops. Some properties o the Bryant Schneider Group o a loop are shown to be true or the Smarandache Bryant Schneider Group o a Smarandache loop. Some interestin and useul cardinality ormulas are also established or a type o inite Smarandache loop. Key Words: Smarandache Bryant Schneider roup, Smarandache loops, Smarandache, -principal isotopes. AMS(2000): 20NO5, 08A05.. Introduction The study o Smarandache loops was initiated by W. B. Vasantha Kandasamy in In her book [6], she deined a Smarandache loop (S-loop) as a loop with at least a subloop which orms a subroup under the binary operation o the loop. For more on loops and their properties, readers should check [4], [3], [5], [7], [6] and [6]. In her book, she introduced over 75 Smarandache concepts in loops but the concept Smarandache Bryant Schneider Group which is to be studied here or the irst time is not amon. In her irst paper [7], she introduced some types o Smarandache loops. The present author has contributed to the study o S-quasiroups and S-loops in [9], [0] and [] while Muktibodh [3] did a study on the irst. Robinson [5] introduced the idea o Bryant-Schneider roup o a loop because its importance and motivation stem rom the work o Bryant and Schneider [4]. Since the advent o the Bryant-Schneider roup, some studies by Adeniran [], [2] and Chiboka [6] have been done on it relative to CC-loops, C-loops and extra loops ater Robinson [5] studied the Bryant-Schneider roup o a Bol loop. The judicious use o it was earlier predicted by Robinson [5]. As mentioned in [Section 5, Robinson [5]], the Bryant-Schneider roup o a loop is extremely useul in investiatin isotopy-isomorphy condition(s) in loops. In this study, the concept o Smarandache Bryant Schneider Group o a Smarandache Received March 6, Accepted April 2, 2008.

2 52 Tèmító. pé. Gbó. láhàn Jaíyéọlá loop is introduced. Relationship(s) between the Bryant Schneider Group and the Smarandache Bryant Schneider Group o an S-loop are discovered and the later is ound to be useul in indin Smarandache isotopy-isomorphy condition(s) in S-loops just like the ormal is useul in indin isotopy-isomorphy condition(s) in loops. Some properties o the Bryant Schneider Group o a loop are shown to be true or the Smarandache Bryant Schneider Group o a Smarandache loop. Some interestin and useul cardinality ormulas are also established or a type o inite Smarandache loop. But irst, we state some important deinitions. 2. Deinitions and Notations Deinition 2. Let L be a non-empty set. Deine a binary operation ( ) on L : I x y L or x, y L, (L, ) is called a roupoid. I the system o equations ; a x = b and y a = b have unique solutions or x and y respectively, then (L, ) is called a quasiroup. Furthermore, i there exists a unique element e L called the identity element such that or x L, x e = e x = x, (L, ) is called a loop. Furthermore, i there exist at least a non-empty subset M o L such that (M, ) is a non-trivial subroup o (L, ), then L is called a Smarandache loop(s-loop) with Smarandache subroup(s-subroup) M. The set SY M(L, ) = SY M(L) o all bijections in a loop (L, ) orms a roup called the permutation(symmetric) roup o the loop (L, ). The triple (U, V, W) such that U, V, W SY M(L, ) is called an autotopism o L i and only i xu yv = (x y)w x, y L. The roup o autotopisms(under componentwise multiplication [4]) o L is denoted by AUT(L, ). I U = V = W, then the roup AUM(L, ) = AUM(L) ormed by such U s is called the automorphism roup o (L, ). I L is an S-loop with an arbitrary S-subroup H, then the roup SSY M(L, ) = SSY M(L) ormed by all θ SY M(L) such that hθ H h H is called the Smarandache permutation(symmetric) roup o L. Hence, the roup SA(L, ) = SA(L) ormed by all θ SSY M(L) AU M(L) is called the Smarandache automorphism roup o L. Let (G, ) be a loop. The bijection L x : G G deined as yl x = x y, x, y G is called a let translation(multiplication) o G while the bijection R x : G G deined as yr x = y x, x, y G is called a riht translation(multiplication) o G. Deinition 2.2(Robinson [5]) Let (G, ) be a loop. A mappin θ SY M(G, ) is a special map or G means that there exist, G so that (θr, θl, θ) AUT(G, ). Deinition 2.3 Let (G, ) be a Smarandache loop with S-subroup (H, ). A mappin θ SSY M(G, ) is a Smarandache special map(s-special map) or G i and only i there exist, H such that (θr, θ) AUT(G, )., θl Deinition 2.4(Robinson [5]) Let the set BS(G, ) = {θ SY M(G, ) :, G (θr, θl, θ) AUT(G, )} i.e the set o all special maps in a loop, then BS(G, ) SY M(G, ) is called the Bryant- Schneider roup o the loop (G, ).

3 On Smarandache Bryant Schneider Group o a Smarandache Loop 53 Deinition 2.5 Let the set SBS(G, ) = {θ SSY M(G, ) : there exist, H (θr, θl, θ) AUT(G, )} i.e the set o all S-special maps in a S-loop, then SBS(G, ) is called the Smarandache Bryant- Schneider roup(sbs roup) o the S-loop (G, ) with S-subroup H i SBS(G, ) SY M(G, ). Deinition 2.6 The triple φ = (R, L, I) is called an, -principal isotopism o a loop (G, ) onto a loop (G, ) i and only i x y = xr yl, x, y G or x y = xr yl, x, y G. and are called translation elements o G or at times written in the pair orm (, ), while (G, ) is called an, -principal isotope o (G, ). On the other hand, (G, ) is called a Smarandache, -principal isotope o (G, ) i or some, S, xr yl = (x y) x, y G where (S, ) is a S-subroup o (G, ). In these cases, and are called Smarandache elements(s-elements). Let (L, ) and (G, ) be S-loops with S-subroups L and G respectively such that xa G, x L, where A : (L, ) (G, ). Then the mappin A is called a Smarandache isomorphism i (L, ) = (G, ), hence we write (L, ) (G, ). An S-loop (L, ) is called a G-Smarandache loop(gs-loop) i and only i (L, ) (G, ) or all S-loop isotopes (G, ) o (L, ). Deinition 2.7 Let (G, ) be a Smarandache loop with an S-subroup H. { Ω(G, ) = (θr, θl }., θ) AUT(G, ) or some, H : hθ H, h H 3. Main Results 3. Smarandache Bryant Schneider Group Theorem 3. Let (G, ) be a Smarandache loop. SBS(G, ) BS(G, ). Proo Let (G, ) be an S-loop with S-subroup H. Comparin Deinitions 2.4 and 2.5, it can easily be observed that SBS(G, ) BS(G, ). The case SBS(G, ) BS(G, ) is possible when G = H where H is the S-subroup o G but this will be a contradiction since G is an S-loop. Identity. I I is the identity mappin on G, then hi = h H, h H and there exists e H where e is the identity element in G such that (IRe, IL e, I) = (I, I, I) AUT(G, ). So, I SBS(G, ). Thus SBS(G, ) is non-empty. Closure and Inverse. Let α, β SBS(G, ). Then there exist,, 2, 2 H such that

4 54 Tèmító. pé. Gbó. láhàn Jaíyéọlá A = (αr, α), B = (βr 2, βl 2, β) AUT(G, ). AB = (αr, α)(r 2 β, L 2 β, β ) = (αr R 2 β L 2 β, αβ ) AUT(G, ). Let δ = βr R 2 β and γ = βl L 2 β. Then, (αβ δ, αβ γ, αβ ) AUT(G, ) (xαβ δ) (yαβ γ) = (x y)αβ x, y G. Puttin y = e and replacin x by xβα, we have (xδ) (eαβ γ) = x or all x G. Similarly, puttin x = e and replacin y by yβα, we have (eαβ δ) (yγ) = y or all y G. Thence, xδr (eαβ γ) = x and yγl (eαβ δ) = y which implies that δ = R (eαβ γ) and γ = L (eαβ δ). Thus, since = eαβ γ, = eαβ δ H then AB = (αβ R, αβ L, αβ ) AUT(G, ) αβ SBS(G, ). Thereore, SBS(G, ) BS(G, ). Corollary 3. Let (G, ) be a Smarandache loop. Then, SBS(G, ) SSY M(G, ) SY M(G, ). Hence, SBS(G, ) is the Smarandache Bryant-Schneider roup(sbs roup) o the S-loop (G, ). Proo Althouh the act that SBS(G, ) SY M(G, ) ollows rom Theorem 3. and the act in [Theorem, [5]] that BS(G, ) SY M(G, ). Nevertheless, it can also be traced rom the acts that SBS(G, ) SSY M(G, ) and SSY M(G, ) SY M(G, ). It is easy to see that SSY M(G, ) SY M(G, ) and that SBS(G, ) SSY M(G, ) while the trivial cases SSY M(G, ) SY M(G, ) and SBS(G, ) SSY M(G, ) will contradict the act that G is an S-loop because these two are possible i the S-subroup H is G. Reasonin throuh the axioms o a roup, it is easy to show that SSY M(G, ) SY M(G, ). By usin the same steps in Theorem 3., it will be seen that SBS(G, ) SSY M(G, ). 3.2 The SBS Group o a Smarandache, -principal isotope Theorem 3.2 Let (G, ) be a S-loop with a Smarandache, -principal isotope (G, ). Then, (G, ) is an S-loop. Proo Let (G, ) be an S-loop, then there exist an S-subroup (H, ) o G. I (G, ) is a Smarandache, -principal isotope o (G, ), then x y = xr yl, x, y G which implies x y = xr yl, x, y G

5 On Smarandache Bryant Schneider Group o a Smarandache Loop 55 where, H. So h h 2 = h R h 2 L, h, h 2 H or some, H. Let us now consider the set H under the operation. That is the pair (H, ). Groupoid. Since, H, then by the deinition h h 2 = h R H, h, h 2 H since (H, ) is a roupoid. Thus, (H, ) is a roupoid. h 2 L, h h 2 Quasiroup. With the deinition h h 2 = h R h 2 L, h, h 2 H, it is clear that (H, ) is a quasiroup since (H, ) is a quasiroup. Loop. It can easily be seen that is an identity element in (H, ). So, (H, ) is a loop. Group. Since (H, ) is a associative, it is easy to show that (H, ) is associative. Hence, (H, ) is an S-subroup in (G, ) since the latter is a loop(a quasiroup with identity element ). Thereore, (G, ) is an S-loop. Theorem 3.3 Let (G, ) be a Smarandache loop with an S-subroup (H, ). A mappin θ SY M(G, ) is a S-special map i and only i θ is an S-isomorphism o (G, ) onto some Smarandache, -principal isotopes (G, ) where, H. Proo By Deinition 2.3, a mappin θ SSY M(G) is a S-special map implies there exist, H such that (θr, θl, θ) AUT(G, ). It can be observed that (θr, θl, θ) = (θ, θ, θ)(r, L, I) AUT(G, ). But since (R, L, I) : (G, ) (G, ) then or (θr, θl, θ) AUT(G, ) we must have (θ, θ, θ) : (G, ) (G, ) which means (G, ) θ θ = (G, ), hence (G, ) (G, ) because (H, )θ = (H, ). (R, L, I) : (G, ) (G, ) is an, -principal isotopism so (G, ) is a Smarandache, -principal isotope o (G, ) by Theorem 3.2. Conversely, i θ is an S-isomorphism o (G, ) onto some Smarandache, -principal isotopes (G, ) where, H such that (H, ) is a S-subroup o (G, ) means (θ, θ, θ) : (G, ) (G, ), (R, L, I) : (G, ) (G, ) which implies (R, L, I) : (G, ) (G, ) and (H, )θ = (H, ). Thus, (θr, θl, θ) AUT(G, ). Thereore, θ is a S-special map because, H. Corollary 3.2 Let (G, ) be a Smarandache loop with an S-subroup (H, ). A mappin θ SBS(G, ) i and only i θ is an S-isomorphism o (G, ) onto some Smarandache, -principal isotopes (G, ) such that, H where (H, ) is an S-subroup o (G, ). Proo This ollows rom Deinition 2.5 and Theorem 3.3. Theorem 3.4 Let (G, ) and (G, ) be S-loops. (G, ) is a Smarandache, -principal isotope o (G, ) i and only i (G, ) is a Smarandache, -principal isotope o (G, ). Proo Let (G, ) and (G, ) be S-loops such that i (H, ) is an S-subroup in (G, ), then (H, ) is an S-subroup o (G, ). The let and riht translation maps relative to an element x

6 56 Tèmító. pé. Gbó. láhàn Jaíyéọlá in (G, ) shall be denoted by L x and R x respectively. I (G, ) is a Smarandache, -principal isotope o (G, ) then, x y = xr yl, x, y G or some, H. Thus, xr y = xr R yl and yl x = yl L xr x, y G and we have R y = R R yl and L x = L L xr, x, y G. So, R y = R R yl and L x = L L xr = x, y G. Puttin y = and x = respectively, we now et R = R R L L L R = L. That is, R = R and L = L or some, H. Recall that x y = xr yl, x, y G x y = xr So usin the last two translation equations, yl, x, y G. x y = xr yl, x, y G the triple (R, L, I) : (G, ) (G, ) = R and L = is a Smarandache, -principal isotopism. Thereore, (G, ) is a Smarandache, -principal isotope o (G, ). The converse is achieved by doin the reverse o the procedure described above. Theorem 3.5 I (G, ) is an S-loop with a Smarandache, -principal isotope (G, ), then SBS(G, ) = SBS(G, ). Proo Let (G, ) be the Smarandache, -principal isotope o the S-loop (G, ) with S- subroup (H, ). By Theorem 3.2, (G, ) is an S-loop with S-subroup (H, ). The let and riht translation maps relative to an element x in (G, ) shall be denoted by L x and R x respectively. Let α SBS(G, ), then there exist, H so that (αr, α) AUT(G, ). Recall that the triple (R, L, I) : (G, ) (G, ) is a Smarandache, -principal isotopism, so x y = xr yl, x, y G and this implies R x = R R xl and L x = L L xr, x G which also implies that R xl = R R x and L xr = L L x, x G which inally ives R x = R R xl and L x = L L xr, x G. Set 2 = αr R and 2 = αl L. Then R 2 = R R αl L L = R R αl, () L 2 = L L αr R R = L L αr, x G. (2) Since, (αr, α) AUT(G, ), then (xαr ) (yαl ) = (x y)α, x, y G. (3)

7 On Smarandache Bryant Schneider Group o a Smarandache Loop 57 Puttin y = and x = separately in the last equation, xαr R (αl ) = xr α and yαl L (αr ) = yl α, x, y G. Thus by applyin () and (2), we now have αr = R αr = R (αl ) αr 2 R and αl = L αl = L (αr ) αl 2 L. (4) We shall now compute (x y)α by (2) and (3) and then see the outcome. (x y)α = (xr yl )α = xr αr yl αl = xr yαl 2 L = xαr 2 yαl 2, x, y G. xαr 2 R Thus, R αr 2 R yl L αl 2 L = (x y)α = xαr 2 yαl 2, x, y G (αr 2 2, α) AUT(G, ) α SBS(G, ). Whence, SBS(G, ) SBS(G, ). Since (G, ) is the Smarandache, -principal isotope o the S-loop (G, ), then by Theorem 3.4, (G, ) is the Smarandache, -principal isotope o (G, ). So ollowin the steps above, it can similarly be shown that SBS(G, ) SBS(G, ). Thereore, the conclusion that SBS(G, ) = SBS(G, ) ollows. 3.3 Cardinality Formulas Theorem 3.6 Let (G, ) be a inite Smarandache loop with n distinct S-subroups. I the SBS roup o (G, ) relative to an S-subroup (H i, ) is denoted by SBS i (G, ), then BS(G, ) = n n SBS i (G, ) [BS(G, ) : SBS i (G, )]. i= Proo Let the n distinct S-subroups o G be denoted by H i, i =, 2, n. Note here that H i H j, i, j =, 2, n. By Theorem 3., SBS i (G, ) BS(G, ), i =, 2, n. Hence, by the Larane s theorem o classical roup theory, BS(G, ) = SBS i (G, ) [BS(G, ) : SBS i (G, )], i =, 2, n. Thus, addin the equation above or all i =, 2, n, we et n n BS(G, ) = SBS i (G, ) [BS(G, ) : SBS i (G, )], i =, 2, n, thence, i= BS(G, ) = n SBS i (G, ) [BS(G, ) : SBS i (G, )]. n i= Theorem 3.7 Let (G, ) be a Smarandache loop. Then, Ω(G, ) AUT(G, ). Proo Let (G, ) be an S-loop with S-subroup H. By Deinition 2.7, it can easily be observed that Ω(G, ) AUT(G, ).

8 58 Tèmító. pé. Gbó. láhàn Jaíyéọlá Identity. I I is the identity mappin on G, then hi = h H, h H and there exists e H where e is the identity element in G such that (IRe, IL e, I) = (I, I, I) AUT(G, ). So, (I, I, I) Ω(G, ). Thus Ω(G, ) is non-empty. Closure and Inverse. Let A, B Ω(G, ). Then there exist α, β SSY M(G, ) and some,, 2, 2 H such that A = (αr, α), B = (βr 2, βl 2, β) AUT(G, ). AB = (αr, α)(r 2 β, L 2 β, β ) = (αr R 2 β L 2 β, αβ ) AUT(G, ). Usin the same techniques or the proo o closure and inverse in Theorem 3. here and by lettin δ = βr R 2 β and γ = βl L 2 β, it can be shown that, AB = (αβ R, αβ L, αβ ) AUT(G, ) where = eαβ γ, = eαβ δ H such that αβ SSY M(G, ) AB Ω(G, ). Thereore, Ω(G, ) AUT(G, ). Theorem 3.8 Let (G, ) be a Smarandache loop with an S-subroup H such that, H and α SBS(G, ). I the mappin Φ : Ω(G, ) SBS(G, ) is deined as Φ : (αr then Φ is an homomorphism., αl, α) α, Proo Let A, B Ω(G, ). Then there exist α, β SSY M(G, ) and some,, 2, 2 H such that A = (αr, α), B = (βr 2, βl 2, β) AUT(G, ). Φ(AB) = Φ[(αR, α)(βr 2, βl 2, β)] = Φ(αR βr 2 βl 2, αβ). It will be ood i this can be written as; Φ(AB) = Φ(αβδ, αβγ, αβ) such that hαβ H h H and δ = R, γ = L or some, H. This is done as ollows. I (αr βr 2 βl 2, αβ) = (αβδ, αβγ, αβ) AUT(G, ), then, xαβδ yαβγ = (x y)αβ, x, y G. Put y = e and replace x by xβ α then xδ eαβγ = x δ = R eαβγ. Similarly, put x = e and replace y by yβ α. Then, eαβδ yγ = y γ = L eαβδ. So, Φ(AB) = (αβr eαβγ, αβl eαβδ, αβ) = αβ = Φ(αR, α)φ(βr 2, βl 2, β) = Φ(A)Φ(B).

9 On Smarandache Bryant Schneider Group o a Smarandache Loop 59 Thereore, Φ is an homomorphism. Theorem 3.9 Let (G, ) be a Smarandache loop with an S-subroup H such that, H and α SSY M(G, ). I the mappin then, Φ : Ω(G, ) SBS(G, ) is deined as Φ : (αr A = (αr, αl, αl, α) kerφ i and only i α, α) α is the identity map on G, is the identity element o (G, ) and N µ (G, ) the middle nucleus o (G, ). Proo For the necessity, kerφ = {A Ω(G, ) : Φ(A) = I}. So, i A = (αr, α) kerφ, then Φ(A) = α = I. Thus, A = (R, L, I) AUT(G, ) x y = xr Replace x by xr and y by yl in (5) to et yl, x, y G. (5) x y = x y, x, y G. (6) Puttin x = y = e in (6), we et = e. Replace y by yl Put x = e in (7), then we have yl in (6) to et x yl = x y, x, y G. (7) = y, y G and so (7) now becomes x (y) = x y, x, y G N µ (G, ). For the suiciency, let α be the identity map on G, the identity element o (G, ) and N µ (G, ). Thus, = = e =. Thus, = e. Then also, y = y = y y G which results into yl xr α yl α = xr Thus, Φ(A) = Φ(αR = y y G. Thus, it can be seen that xαr yαl yl = xr y = (xr )y = xr, α) = Φ(R, L, I) = I A kerφ. = xr yl R y = x y = (x y)α, x, y G. = Theorem 3.0 Let (G, ) be a Smarandache loop with an S-subroup H such that, H and α SSY M(G, ). I the mappin Φ : Ω(G, ) SBS(G, ) is deined as Φ : (αr, α) α then, N µ (G, ) = kerφ and Ω(G, ) = SBS(G, ) N µ (G, ). Proo Let the identity map on G be I. Usin Theorem 3.9, i θ = (R, L, I), N µ (G, ) then, θ : N µ (G, ) kerφ. θ is easily seen to be a bijection, hence N µ (G, ) = kerφ.

10 60 Tèmító. pé. Gbó. láhàn Jaíyéọlá Since Φ is an homomorphism by Theorem 3.8, then by the irst isomorphism theorem in classical roup theory, Ω(G, )/ kerφ = ImΦ. Φ is clearly onto, so ImΦ = SBS(G, ), so that Ω(G, )/ kerφ = SBS(G, ). Thus, Ω(G, )/ kerφ = SBS(G, ). By Larane s theorem, Ω(G, ) = kerφ Ω(G, )/ kerφ, so, Ω(G, ) = kerφ SBS(G, ), Ω(G, ) = N µ (G, ) SBS(G, ). Theorem 3. Let (G, ) be a Smarandache loop with an S-subroup H. I Θ(G, ) = {(, ) H H : (G, ) (G, ) or (G, ) the Smarandache principal, isotope o (G, )}, then Ω(G, ) = Θ(G, ) SA(G, ). Proo Let A, B Ω(G, ). Then there exist α, β SSY M(G, ) and some,, 2, 2 H such that A = (αr, α), B = (βr 2, βl 2, β) AUT(G, ). Deine a relation on Ω(G, ) such that A B = 2 and = 2. It is very easy to show that is an equivalence relation on Ω(G, ). It can easily be seen that the equivalence class [A] o A Ω(G, ) is the inverse imae o the mappin Ψ : Ω(G, ) Θ(G, ) deined as Ψ : (αr, α) (, ). I A, B Ω(G, ) then Ψ(A) = Ψ(B) i and only i (, ) = ( 2, 2 ) so, = 2 and = 2. Since Ω(G, ) AUT(G, ) by Theorem 3.7, then AB = (αr, α)(βr 2, βl 2, β) = (αr R 2 β L 2 β, αβ ) = (αβ, αβ, αβ ) AUT(G, ) αβ SA(G, ). So, A B αβ SA(G, ) and (, ) = ( 2, 2 ). Whence, [A] = SA(G, ). But each A = (αr, α) Ω(G, ) is determined by some, H. So since the set {[A] : A Ω(G, )} o all equivalence classes partitions Ω(G, ) by the undamental theorem o equivalence relation, Ω(G, ) =, H [A] =, H SA(G, ) = Θ(G, ) SA(G, ). Thereore, Ω(G, ) = Θ(G, ) SA(G, ). Theorem 3.2 Let (G, ) be a inite Smarandache loop with a inite S-subroup H. (G, ) is S-isomorphic to all its S-loop S-isotopes i and only i (H, ) 2 SA(G, ) = SBS(G, ) N µ (G, ).

11 On Smarandache Bryant Schneider Group o a Smarandache Loop 6 Proo As shown in [Corollary 5.2, [2]], an S-loop is S-isomorphic to all its S-loop S-isotopes i and only i it is S-isomorphic to all its Smarandache, principal isotopes. This will happen i and only i H H = Θ(G, ) where Θ(G, ) is as deined in Theorem 3.. Since Θ(G, ) H H then it is easy to see that or a inite Smarandache loop with a inite S-subroup H, H H = Θ(G, ) i and only i H 2 = Θ(G, ). So the proo is complete by Theorems Corollary 3.3 Let (G, ) be a inite Smarandache loop with a inite S-subroup H. (G, ) is a GS-loop i and only i (H, ) 2 SA(G, ) = SBS(G, ) N µ (G, ). Proo This ollows by the deinition o a GS-loop and Theorem 3.2. Lemma 3. Let (G, ) be a inite GS-loop with a inite S-subroup H and a middle nucleus N µ (G, ). SBS(G, ) (H, ) = N µ (G, ) (H, ) = SA(G, ). Proo From Corollary 3.3, ()I (H, ) = N µ (G, ), then (H, ) 2 SA(G, ) = SBS(G, ) N µ (G, ). (H, ) SA(G, ) = SBS(G, ) = (H, ) = SBS(G, ) SA(G, ). (2)I (H, ) = SBS(G, ) SA(G, ), then (H, ) SA(G, ) = SBS(G, ). Hence, multiplyin both sides by (H, ), So that (H, ) 2 SA(G, ) = SBS(G, ) (H, ). SBS(G, ) N µ (G, ) = SBS(G, ) (H, ) = (H, ) = N µ (G, ). Corollary 3.4 Let (G, ) be a inite GS-loop with a inite S-subroup H. I N µ (G, ), then, SBS(G, ) n SBS(G, ) (H, ) =. Hence, (G, ) = or some n. SA(G, ) SA(G, ) Proo By hypothesis, {e} H G. In a loop, N µ (G, ) is a subroup, hence i N µ (G, ), then, we can take (H, ) = N µ (G, ). So that (H, ) = N µ (G, ). Thus by Lemma 3., (H, ) = SBS(G, ) SA(G, ).

12 62 Tèmító. pé. Gbó. láhàn Jaíyéọlá As shown in [Section.3, [8]], a loop L obeys the Larane s theorem relative to a subloop H i and only i H(hx) = Hx or all x L and or all h H. This condition is obeyed by N µ (G, ), hence there exists n N such that (H, ) (G, ) = (G, ) = SBS(G, ) SA(G, ) n SBS(G, ). SA(G, ) (G, ) = But i n =, then (G, ) = (H, ) = (G, ) = (H, ) hence (G, ) is a roup which is a contradiction to the act that (G, ) is an S-loop. Thereore, (G, ) = n SBS(G, ) SA(G, ) or some natural numbers n. Reerences [] J. O. Adeniran (2002), The study o properties o certain class o loops via their Bryant- Schneider roup, Ph.D thesis, University o Ariculture, Abeokuta. [2] J. O. Adeniran (2002), More on the Bryant-Schneider roup o a conjuacy closed loop. Proc. Janjeon Math. Soc. 5, no., [3] R. H. Bruck (966), A survey o binary systems, Spriner-Verla, Berlin-Göttinen-Heidelber, 85pp. [4] B. F. Bryant and H. Schneider (966), Principal loop-isotopes o quasiroups, Canad. J. Math. 8, [5] O. Chein, H. O. Pluelder and J. D. H. Smith (990), Quasiroups and loops : Theory and applications, Heldermann Verla, 568pp. [6] V. O. Chiboka (996), The Bryant-Schneider roup o an extra loop, Collection o Scientiic papers o the Faculty o Science, Kraujevac, 8, [7] J. Déne and A. D. Keedwell (974), Latin squares and their applications, the Enlish University press Lts, 549pp. [8] E. G. Goodaire, E. Jespers and C. P. Milies (996), Alternative loop rins, NHMS(84), Elsevier, 387pp. [9] T. G. Jaíyéọlá (2006), An holomorphic study o the Smarandache concept in loops, Scientia Mana Journal, Vol.2, No., -8. [0] T. G. Jaíyéọlá (2006), Parastrophic invariance o Smarandache quasiroups, Scientia Mana Journal, Vol.2, No.3, [] T. G. Jaíyéọlá (2006), On the universality o some Smarandache loops o Bol-Mouan type, Scientia Mana Journal, Vol.2, No.4, [2] T. G. Jaíyéọlá (2008), Smarandache isotopy theory o Smarandache: quasiroups and loops, Proceedins o the 4 th International Conerence on Number Theory and Smarandache Problems, Scientia Mana Journal. Vol.4, No.,

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