Consequences of Saito's results

It is convenient to begin with a slight rephrasing of part of Saito's main theorem.

Theorem 2.1   Let 54#54 and 55#55 If 56#56 then 57#57.

Proof: Suppose first that 58#58Then by Saito's theorem,

59#59

and so 57#57. If 60#60 then 51#51 and so

61#61

Thus 57#57. 32#32

Theorem 2.2   Let 54#54 and let 62#62. If n - d is odd, then

63#63

if n - d is even, then

64#64

Proof: We maximize 3#3 by having as many cyclic components and as few fixed points as possible, subject to the condition that 62#62. The d elements of 65#65 cannot appear in cyclic components, and must have at least one image, which again cannot be part of a cyclic component, so the maximum number of elements available to make cyclic components is n - d - 1. If n - d is odd, we can form 66#66 cyclic components of order 2. The remaining d + 1 elements must then all map to one of their number. This gives 67#67, 68#68, and hence

69#69

Within the given constraints this is the largest 3#3 that can be obtained. If 70#70 it can also be obtained using n - d - 3elements to form 71#71 cyclic components and forming the remaining d + 3 elements into a single ``leftover'' component of one of the types illustrated in Figure 2.

 

Figure: Leftover Components

72#72

Yet another way (if 73#73) is to form 74#74 cyclic components of order 2, one cyclic component of order 3 and to form the remaining d + 2 elements into a single ``leftover'' component of the type illustrated in Figure 3.

 

Figure: Single Leftover Component

72#72

All these approaches lead to the same value 75#75 for 3#3.

Suppose now that n - d is even. Then the unique way of forming an element with largest possible 3#3 is to form n - d - 2 elements into 66#66 cyclic components of order 2 and to make the remaining d + 2 elements into a single component of the type illustrated in Figure 3. This gives 76#76, 77#77, and hence

78#78

It is convenient now to write 79#79 (the rank of 1#1) for 80#80, noting that 81#81. In [4] it was shown that if 82#82 then 83#83. The next theorem is a more general result of the same kind.

Theorem 2.3   Let 54#54 and let 84#84. If 79#79 is even and

85#85

then 57#57. If 79#79 is odd and

86#86

then 57#57.

Proof: If 79#79 ( 87#87 ) is even and if

85#85

then

88#88

which can be rearranged to give

89#89

Hence by Theorems  2.1 and  2.2,

90#90

and so 57#57. The case where 79#79 is odd is similar. 32#32

For p = 3 we see that 83#83 if

91#91

which is a weaker result than that obtained in [4]. On the other hand Howie, Robertson and Schein privately conjectured that 92#92 if 93#93 and the theorem above gives the stronger result that 92#92 if

94#94