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identical with the second normalizer for all subloops.
10. Characterize those loops L for which the first and the second normalizer are
distinct for all subloops of L.
11. Does there exists a A-loop, which is not Moufang?
12. Does there exist a C-loop, which is not a S-loop? or is every C-loop a S-loop?
13. Characterize those loops L (where L is not a loop in the class L with n a
n
prime), which are S-subgroup loop.
14. Does there exist a loop L that has normal subloops but they are not S-normal
subloops?
15. Can the class of loops L for any odd n, n > 3 have a subgroup whose order is
n
greater than four?
16. Can we ever have a class of loops, which contains the group Sn as its subgroup?
(Does not include the S-mixed direct product of loops).
17. Can we have an extension of Cayleys theorem for at least S-loops.(Hint: If the
solution to the above problem is true certainly a solution to this problem exists
or we have Cayleys theorem to be true for S-loops).
18. Does there exists a S-strongly commutative loop, which is not a S-strongly
cyclic loop?
19. Characterize those loops L which have only one S-commutator subloop.
20. Characterize those loops L that has always the S-commutator subloop to be
coincident with the commutator subloop. (Not loops from the class L ).
n
21. Characterize those loops L, which has n-S-subloop and n distinct S-
commutator subloops, associated with them. (Is this possible?)
22. Characterize those loops L that has many distinct S-subloops but one and only
one S-commutator subloop.
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23. Characterize those loops L that has always the S-associator subloop to be
coincident with the associator subloop of L. (Not loops from the class L as it
n
has been already studied.)
24. Characterize those loops L, which has n-S-subloops and n distinct S-associator
subloops, associated with them.
25. Characterize those loops L that has one and only one S-associator subloop (L
has S-subloops).
26. Characterize those loops L in which LA = PA(LS) = SPA(LS) = A(L).
27. Can loop L (m) where n is not a prime have proper subloops which are
n
i. Moufang loops?
ii. Bruck loops?
iii. Bol loops?
(This will in turn prove L (m) when n is not prime is a S-Moufang loop, S-Bol
n
loop and S-Bruck loop)
28. Can we say every Smarandache strong Moufang triple (Smarandache strong
Bol triple or Smarandache strong Bruck triple) generate a Moufang subloop
(Bol subloop or Bruck subloop) which is a S-subloop of the given loop.
29. Characterize those loops L, which are not Moufang but in which every S-
subloop is a Moufang loop that is L is a S-strongly Moufang loop.
30. A similar problem in case of
i. non-Bol loops which are S-strongly Bol loops.
ii. non-Bruck loops which are S-strong Bruck loops.
iii. non-WIP loops which are S-strong WIP loops.
iv. non-diasscoiative loops which are S-strong diassociative loops.
v. non-power associative loops which are S-strong power associative
loops.
Note: If a loop L has only one S-loop and no other S-subloop which has the
stipulated property L becomes a S-strongly loop having that property.
31. Characterize those loops which have many S-subloops still
i. S-Moufang centre is unique.
ii. S-nucleus SN»SNµSNÁ is unique.
iii. S-centre is unique.
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iv. S-first normalizer is unique.
v. S-second normalizer is unique.
32. Characterize those loops which has several S-subloops having distinct S-
Moufang centre, S-centre, S-nucleus, S-first and S-second normalizer.
33. Characterize those loops for which every S-subloop has SN1 = SN2.
34. Characterize those loops for which every distinct pair of S-subloops SN1 `" SN2.
35. Does there exist examples of S-loops of prime order in which every subgroup
is a normal subgroup?
36. Characterize those loops L in which every subgroup is a normal subgroup.
37. Obtain loops L of odd order not got from the S-mixed direct product of loops
but which satisfy S-Lagrange criteria.
i. Does there exist such loops?
ii. Can you characterize such loops of odd order.
38. Characterize those loops which satisfy S-Sylow criteria or equivalently those
loops, which do not satisfy S-Sylow criteria.
39. Suppose a loop L satisfies S-Lagrange's criteria can we say L satisfies S-Sylow
criteria? Characterize those loops, which satisfy both!
40. For any S-loop L when will every subgroup A give
n
i. = L (Axi `" Axj Axi )" Axj = Æ)
UAxi
i=1
n
ii. A = L; (xi " L; xiA `" xjA, xiA )" xjA = Æ)
Ux i
i=1
iii. Can the same elements {x1, x2, x3, & , x } ‚" L serves the propose?
n
Characterize those S-loops for which i, ii and iii is true.
41. Prove for all non-commutative loops L (m) " L , n a prime for every subgroup
n n
Ai = {e, i}, i = 1, 2, & , n we have the S-right coset decompositions L (m) =
n
n+1
n+1
2 2
xi (Axi )" Axk = Æ, i `" k), L (m) = y (Ayi )" Ayk = Æ, i `" k)
UA j n UA j i
i=1 i=1
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where the set X = {xi, & , x } and Y = {y1, y2, & , y } are such that X
n+1 n+1
2 2
*" Y = L (m) and X )" Y = Æ.
n
42. Study problem 41 in case S-left coset representation for the same class of
loops.
n +1
43. What happens to the coset representation for the loop Lnëø öø which is the
ìø ÷ø
2
íø øø
only commutative loop in the class L ? Can we have problem 42 and 43 to be
n
true? If so illustrate with examples and give characterization theorem for such
loops.
44. Let L be a S-loop of odd order n. Study the coset representation when
i. n is a prime.
ii. n is an odd prime.
45. Let L (m) " L . Suppose B is a subgroup of order greater than 2. When will
n n
i. L (m) = , xi " L (m) (with Bxi )" Bxj = Æ, if xi `" xj)
n UBxi n
ii. L (m) = B , xi " L (m) (with xiB )" xjB = Æ, if xi `" xj)
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