VI 



TABLE OF CONTENTS. 



CHAPTER IV. 

 Miscellaneous properties of Galois Fields. 



Page 



61 62. Squares and not- squares 44 



63. Number of w th powers in a field; extraction of roots . . 45 



6467. Number of sets of solutions of certain quadratics .... 4648 



6871. Additive -groups and their multiplier Galois Fields . . . 4951 



72. Condition for linear independence of marks with respect 



to an included field 52 



73. Conjugacy of marks with respect to an included field . . 52 



74. Newton's identities for sums of powers of the roots of an 

 equation belonging to a Galois Field 5354 



CHAPTER V. 



Analytic representation of substitutions on the marks 

 of a Galois Field. 



76 78. Definitions. Representation of a given substitution . . . 54 55 



7983. Special functions suitable to represent substitutions . . . 5659 



84. Necessary and sufficient conditions for a substitution quantic 5960 



85 89. Applications of preceding theorem. Reduced form . . . 61 63 



90. Table of all substitution quantics of degree < 6 . . . . 6364 



91 94. Betti-Mathieu Group. Certain of its subgroups .... 64 68 



95. Identity of Betti-Mathieu Group in the GF[pnm] with 

 Jordan's linear homogeneous group in the GF[p~] on m indices 69 70 



96. Exercises TO 71 



SECOND PAET, 



THEORY OF LINEAR GROUPS IN A GALOIS FIELD. 



CHAPTER I. 

 General linear homogeneous group. 



9798. Two definitions of the group . . .".".'.' 7577 



99100. Order and generators '..'."..'. 7779 



101 102. Transformation of indices. Invariance of characteristic 



determinant 80 81 



103 107. Factors of composition of the linear homogeneous group . 81 86 



108 109. Linear fractional group. Isomorphic permutation group . 87 88 



CHAPTER H. 

 The Abelian linear group. 



110 112. Conditions for Abelian substitutions. Inverse substitution . 8.9 91 



114 115. Generators and order of Abelian group 92 94 



116 119. Factors of composition of the Abelian linear group . . . 94 100 



120121. Conjugacy of operators of period two of the Abelian group 100 105 



122 123. Operators of period two in the quotient -group J.(2w,jp) 105 109 



