Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

SUBJECT INDEX

385

Perfect group, 69
Period of an element of a group, 53
Permutations and substitutions, 36
~-subgroup, 71
P-isomorphisms, 134
Positive and negative substitutions, 16
Power of a group, 32
Primary groups, 118
Prime-power groups, 118
Primitive and' imprimitive substitution groups, 38
Product of two substitutions, 2
Quadratic group, 65
Quaternion group, 62
Quotient group, 34, 66
Rank of an abelian group, 92
Relative and intrinsic properties of
operators, 159
Regular substitution group, 35
Representation of a group as a regular
substitution group, 63
- - an abstract group as a transitive
substitution group, 81
Right co-sets, 66
Roots of operators of an abelian group,
114
Self-conjugate subgroup, 21
Series of composition,. 177.
Set of independent generators of a
group, 9, 90, 127

Similar substitutions and similar groups,
21
Simple group, 43
- isomorphisms.and simply isomorphic
groups, 33, 73, 95
Simplicity of the alternating group, 43
Solvable group, 174
Subgroup, 3
Subgroups and quotient groups of an
abelian group, 99
Substitution and substitution group,
1,2
- groups of degree five, 45
Substitutions commutative with a
given substitution, 19
Sylow's theorem, 27
Sylow subgroups, 27, 181
Symmetric group, 1, 3, 166
Systems of imprimitivity, 38
Tetrahedral group, 147, 152
Totient of a number, 11
Totitives of a number, 95
Transform of a substitution and of a
substitution group, 20, 57
Transitive constituent of an intransitive group, 33
- substitution group, 31
Transitivity of the symmetric group, 40
Transposition, 16
Vierergruppe, 65
~T II

Abelian groups, canonical form of, 213
Algebraic integer,.241
Canonical form, 196; theorems on,
212, 213
Change of variables, 203
Characteristic and characteristic equation, 205
Collineations and collineation-groups,
198
Conjugate-imaginary groups, 209

Determinant of a linear transformation, 194; theorems on the determinants of the transformations -belonging to a finite group, 196, 200,
202
Dihedral group, 220, 225
Diophantine equation, 227
Equivalent groups; 262
Group-matrix, theorem on, 268