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

TABLE OF CONTENTS xiii
SECTION PAGE
44. Characteristic sub-groups of an Abelian group......................... 109
E xercises........................................................ 112
45. Non-Abelian groups in which every sub-group is Abelian.............. 112
Exercises........................................................ 114
46. Roots of the operators of an Abelian group........................... 114
47. Hamilton groups........................................ 115
Exercises........................................................ 117
CHAPTER V
GROUPS WHOSE ORDERS ARE POWERS OF PRIME NUMBERS
48. Introduction................................1........ 118
49. Invariant Abelian sub-groups....................................... 120
Exercises........................................... 122
50. Number of sub-groups in a group of order ptm....................... 123
E xercises........................................................ 127
51. Number of non-cyclic subgroups in a group of order p", p>2.......... 128
52. Number of non-cyclic subgroups in a group of order 2................. 129
Exercises........................................................ 133
53. Some properties of the group of isomorphisms of a group of order pt..... 134
Exercises....................................................... 137
54. Maximum order of a Sylow subgroup in the group of isomorphisms of a
group of order pm............................................... 137
55. Construction of all the possible groups of order pm................. 138
E xercises.................................................. 141
CHAPTER VI
GROUPS HAVING SIMPLE ABSTRACT DEFINITIONS
56. Groups generated by two operators having a common square........... 143
Exercises................................................. 147
57. Groups of the regular polyhedrons.................................. 147
58. Group of the regular icosahedron............................... 150
Exercises...................................................... 151
59. Generalizations of the group of the regular tetrahedron.............. 152
60. Generalizations of the octahedron group............................. 154
Exercises........................................................ 158
CHAPTER VII
ISOMORPHISMS
61. Relative and intrinsic properties of the operators of a group............ 159
62. Group of isomorphisms as a substitution group...................... 160
63. Groups of isomorphisms of non-Abelian groups...................... 162
IE xercises........................................................ 164
64. Doubly transitive substitution groups of isomorphisms.............. 164