Index

ABEL, quoted, 1
ABELIAN GROUP, (See alsoHOMOGENEOUS LINEAR GROUP.)
    deﬁnition of, 3
    existence of independent generating operations of, 4
    invariance of the orders of a set of independent generating operations of, 5
    of order

p^{m}

and type

(1, 1, \dots, 1)

, 6
        group of isomorphisms of, 7, 8
        holomorph of, 9
        number of distinct ways of choosing a set of independent generating operations of, 10
        number of sub-groups of, whose order is given, 11
    sub-groups of, 12, 13, 14–15
    symbol for, of given type, 16
ALTERNATING GROUP
    deﬁnition of, 17
    group of isomorphisms of, 18
    is simple, except for degree

4

, 19

BOCHERT, quoted, 20
BOLZA, quoted, 21
BURNSIDE, quoted, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31

CAUCHY, quoted, 32
CAYLEY, quoted, 33
CHARACTERISTIC SERIES
    deﬁnition of, 34
    invariance of, 35
    of a group whose order is a power of a prime, 36–37
CHARACTERISTIC SUB-GROUP
    deﬁnition of, 38
    groups with no, are either simple or the direct product of simply isomorphic simple groups, 39
CHIEF COMPOSITION SERIES, or CHIEF SERIES
    deﬁnition of, 40
    invariance of, 41
COLE, quoted, 42, 43
COLE and GLOVER, quoted, 44
COLOUR-GROUPS, 45–46
    examples of, 47
COMPLETE GROUP
    deﬁnition of, 48
    group of isomorphisms of a simple group of composite order is a, 49
    groups which contain a, self-conjugately are direct products, 50
    holomorph of a cyclical group of odd order is a, 51
    holomorph of an Abelian group of order $p^{m}$ and type $(1, 1, \dots, 1)$ is a, 52
    symmetric group is a, except for degree $6$ , 53
COMPOSITE GROUPS
    deﬁnition of, 54
    non-soluble, 55–56
    of even order, 57–58
COMPOSITION FACTORS
    deﬁnition of, 59
COMPOSITION-SERIES
    deﬁnition of, 60
    examples of, 61, 62
    invariance of, 63
CONJUGATE OPERATIONS
    complete set of, 64
    deﬁnition of, 65
CONJUGATE SUB-GROUPS
    complete set of, 66
    deﬁnition of, 67
    operations common to or permutable with a complete set of, form a self-conjugate sub-group, 68

DEDEKIND, quoted, 69
DEFINING RELATIONS of a group
    deﬁnition of, 70
    for groups of genus unity, 71, 72
    for groups of genus zero, 73
    for groups of orders $p^{2}$ , $p^{3}$ , $p^{4}$ , 74, 75, 76
    for groups of order $p^{2} q$ , 77–78
    for groups of order $p q$ , 79
    for groups whose orders contain no square factor, 80
    for the holomorph of a cyclical group, 81, 82
    for the simple group of order $168$ , 83
    for the symmetric group of degree $5$ , 84
    limitation on the number of, when the genus is given, 85
DEGREE of a substitution group
    deﬁnition of, 86
    is a factor of the order, if the group is transitive, 87
DIRECT PRODUCT of two groups
    deﬁnition of, 88
    represented as a transitive group, 89
DIRECT PRODUCT of two simply isomorphic groups of order $n$ represented as a transitive group of degree $n$ , 90, 91
DOUBLY TRANSITIVE GROUPS
    generally contain simple self-conjugate sub-groups, 92
    of degree $n$ and order $n (n - 1)$ , 93–94
    the sub-groups of, which keep two symbols ﬁxed, 95
    with a complete set of triplets, 96
DYCK, quoted, 97, 98, 99, 100, 101, 102, 103, 104
DYCK’S THEOREM that a group of order $n$ can be represented as a regular substitution group of degree $n$ , 105–106

FACTOR GROUPS
    deﬁnition of, 107
    invariance of, 108
    set of, for a given group, 109
FORSYTH, quoted, 110, 111
FRACTIONAL LINEAR GROUP
    analysis of, 112–113
    deﬁnition of, 114
    generalization of, 115, 116
FROBENIUS, quoted, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133
FROBENIUS AND STICKELBERGER, quoted, 134
FROBENIUS’S THEOREM that, if $n$ is a factor of the order of a group, the number of operations of the group whose orders divide $n$ is a multiple of $n$ , 135–136

GALOIS, quoted, 137, 138
GENERAL DISCONTINUOUS GROUP with a ﬁnite number of generating operations, 139
    relation of special groups to, 140, 141
GENUS of a group, deﬁnition of, 142
GIERSTER, quoted, 143
GRAPHICAL REPRESENTATION
    examples of, 144–145, 146, 147–148, 149, 150, 151, 152, 153, 154
    of a cyclical group, 155, 156, 157
    of a general group, 158–159
    of a group of ﬁnite order, 160–161
    of a special group, 162–163
GROUP, (See alsoSUBSTITUTION GROUP.)
    Abelian, 165
    alternating, 166
    complete, 167
    continuous, discontinuous, or mixed, 168, 169
    cyclical, 170
    deﬁning relations of, 171
    deﬁnition of, 172, 173
    dihedral, 174
    fundamental or generating operations of, 175
    general, 176
    genus of a, 177
    group of isomorphisms of a, 178
    holomorph of a, 179
    icosahedral, 180
    multiplication table of, 181, 182
    octohedral, 183
    order of, 184
    quadratic, 185
    simple and composite, 186
    soluble, 187
    special, 188
    symbol for a, 189
    symmetric, 190
    tetrahedral, 191
GROUP OF ISOMORPHISMS
    contains the group of cogredient isomorphisms self-conjugately, 192
    of a cyclical group, 193–194
    of an Abelian group of order $p^{n}$ and type $(1, 1, \dots, 1)$ , 195, 196, 197–198, 199–200
    of doubly transitive groups of degree $p^{n} + 1$ and order $\frac{1}{2} p^{n} (p^{2 n} - 1)$ , 201–202
    of the alternating group, 203
GROUPS
    of genus one, 204–205
    of genus two, 206
    of genus zero, 207–208
GROUPS WHOSE ORDER IS $p^{m}$ , where $p$ is a prime, 209 et seq.
    always contain self-conjugate operations, 210
    case in which there is only one sub-group of a given order, 211–212
    determination of distinct types of orders $p^{2}$ , $p^{3}$ , $p^{4}$ , where $p$ is an odd prime, 213–214
    every sub-group of, is contained self-conjugately in a greater sub-group, 215
    number of sub-groups of given order is congruent to unity, $(mod p)$ , 216
    number of types which contain self-conjugate cyclical sub-groups of order $p^{m - 2}$ , 217–218
    table of distinct types of orders $8$ and $16$ , 219, 220
GROUPS whose orders contain
    no cubed factor, 221
    no squared factor, 222, 223
GROUPS whose sub-groups of order $p^{α}$ are all cyclical, 224, 225

HEFFTER, quoted, 226
HöLDER, quoted, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238
HOLOMORPH
    deﬁnition of, 239
    of a cyclical group, 240–241
    of an Abelian group of order $p^{n}$ and type $(1, 1, \dots, 1)$ , 242
HOMOGENEOUS LINEAR GROUP
    composition series of, 243–244
    deﬁnition of, 245
    generalization of, 246–247
    represented as a transitive substitution group, 248, 249
    simple groups deﬁned by, 250, 251
HURWITZ, quoted, 252, 253, 254

IDENTICAL OPERATION
    deﬁnition of, 255
IMPRIMITIVE GROUPS
    deﬁnition of, 256
    of degree $6$ , 257, 258
IMPRIMITIVE SELF-CONJUGATE SUB-GROUP of a doubly transitive group, 259, 260
IMPRIMITIVE SYSTEMS
    deﬁnition of, 261
    of a regular group, 262
    of any transitive group, 263
    properties of, 264, 265
INTRANSITIVE GROUPS
    deﬁnition of, 266
    of degree $7$ with transitive sets of $3$ and $4$ , 267, 268
    properties of, 269–270
    transitive sets of symbols in, 271, 272
ISOMORPHISM of general and special groups, 273, 274
ISOMORPHISM OF TWO GROUPS
    general, deﬁnition of, 275
    multiple, deﬁnition of, 276
    simple, deﬁnition of, 277
ISOMORPHISMS of a group with itself
    class of, deﬁnition of, 278
    cogredient and contragredient, deﬁnition of, 279
    deﬁnition of, 280
    limitation on the order of, 281

JORDAN, quoted, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293

KLEIN, quoted, 294, 295, 296
KLEIN and FRICKE, quoted, 297

LIMITATION
on the number of deﬁning relations of a group of given genus, 298
on the order of a group of given genus, 299

MAILLET, quoted, 300
MARGGRAFF, quoted, 301
MASCHKE, quoted, 302
MATHIEU, quoted, 303
MAXIMUM SELF-CONJUGATE SUB-GROUP
    deﬁnition of, 304
MAXIMUM SUB-GROUP, deﬁnition of, 305
MILLER, quoted, 306, 307, 308
MINIMUM SELF-CONJUGATE SUB-GROUP
    deﬁnition of, 309
    is a simple group or the direct product of simply isomorphic simple groups, 310
MOORE, quoted, 311, 312
MULTIPLICATION TABLE of a group, 313, 314
MULTIPLY ISOMORPHIC GROUPS
    deﬁnition of, 315

NETTO, quoted, 316, 317, 318
NUMBER OF SYMBOLS unchanged by all the substitutions of a group, 319, 320

OPERATIONS
    common to or permutable with each of a complete set of conjugate sub-groups form a self-conjugate sub-group, 321
    common to two groups form a group, 322
    of a group, which are permutable with a given operation or sub-group, form a group, 323, 324
ORDER
    of a group, deﬁnition of, 325
    of an operation, deﬁnition of, 326

PERMUTABILITY of an operation with a group
    deﬁnition of, 327
PERMUTABLE GROUPS
    deﬁnition of, 328
PERMUTABLE OPERATIONS
    deﬁnition of, 329
PRIMITIVE GROUPS
    deﬁnition of, 330
    limit to the order of, for a given degree, 331
    of degrees $3$ , $4$ and $5$ , 332
    of degree $6$ , 333, 334
    of degree $7$ , 335–336
    of degree $8$ , 337–338
    of prime degree, 339
    when soluble, have a power of a prime for degree, 340
    with a transitive sub-group of smaller degree, 341, 342
PRIMITIVITY, test of, 343

REGULAR DIVISION of a surface, representation of a group by means of, 344
REPRESENTATION, graphical, (See GRAPHICAL REPRESENTATION.)
REPRESENTATION OF A GROUP
in primitive form, 346
in transitive form; i.e. as a transitive substitution group, 347, 348–349

SELF-CONJUGATE OPERATION
    deﬁnition of, 350
    of a group whose order is the power of a prime, 351
    of a transitive substitution group must be a regular substitution, 352
SELF-CONJUGATE SUB-GROUP
    deﬁnition of, 353
    generated by a complete set of conjugate operations, 354
    of a $k$ -ply transitive group is, in general, $(k - 1)$ -ply transitive, 355
    of a primitive group must be transitive, 356
    of an imprimitive group, 357, 358
SIMPLE GROUPS
    deﬁnition of, 359
    whose orders are the products of not more than $5$ primes, 360–361
    whose orders do not exceed $660$ , 362–363
SIMPLY ISOMORPHIC GROUPS
    deﬁnition of, 364
    said to be of the same type, 365
SOLUBLE GROUPS
    deﬁnition of, 366
    properties of, 367, 368
    special classes of, 369–370
SUB-GROUP, (See alsoCONJUGATE, SELF-CONJUGATE and CHARACTERISTIC SUB-GROUP.)
    deﬁnition of, 372
    order of a, divides order of group, 373
SUBSTITUTION
    circular, 374
    cycles of, 375
    deﬁnition of, 376
    even and odd, 377
    identical, 378
    inverse, 379
    order of, 380
    permutable, 381
    regular, 382
    similar, 383
    symbol for the product of two or more, 384
SUBSTITUTION GROUP
    conjugate substitutions of, are similar, 385
    construction of multiply transitive, 386
    degree of transitive, is a factor of its order, 387
    degree of, deﬁnition of, 388
    doubly transitive, of degree $n$ and order $n (n - 1)$ , 389–390
    limit to the degree of transitivity of, 391
    multiply transitive, deﬁnition of, 392
    order of a $k$ -ply transitive, whose degree is $n$ , is a multiple of $n (n - 1) \dots (n - k + 1)$ , 393
    primitive and imprimitive, deﬁnition of, 394
    quintuply transitive, of degree $12$ , 395
    regular, deﬁnition of, 396
    representation of any group as a regular, 397
    transitive and intransitive, deﬁnition of, 398
    transitive, whose substitutions, except identity, displace all or all but one of the symbols, 399–400
    triply transitive, of degree $n$ and order $n (n - 1) (n - 2)$ , 401
SUBSTITUTION GROUPS whose orders are powers of primes, 402, 403
SUBSTITUTIONS which are permutable
    with a given substitution, 404, 405
    with every substitution of a given group, 406, 407, 408
    with every substitution of a group, whose degree is equal to its order, form a simply isomorphic group, 409
SYLOW, quoted, 410, 411
SYLOW’S THEOREM, 412–413
    extension of, 414
    some direct consequences of, 415–416
SYMBOL
    for a group generated by given operations, 417
    for the product of two or more operations, 418
    for the product of two or more substitutions, 419
     $𝜗 (P)$ and $𝜃 (P)$ , deﬁnition of, 420, 421
SYMMETRIC GROUP
    deﬁnition of, 422
    is a complete group, except for degree $6$ , 423
    of degree $6$ has $12$ simply isomorphic sub-groups of order $5!$ which form two distinct conjugate sets, 424
    of degree $n$ has a single set of conjugate sub-groups of order $(n - 1)!$ except when $n$ is $6$ , 425

TRANSFORMING AN OPERATION
    deﬁnition of, 426
TRANSITIVE GROUP
    deﬁnition of, 427
    number of distinct modes of representing the alternating group of degree $5$ as a, 428, 429
    representation of any group as a, 430–431
TRANSPOSITIONS
    deﬁnition of, 432
    number of, which enter in the representation of a substitution is either always even or always odd, 433
    representation of a substitution by means of, 434
TRIPLY TRANSITIVE groups of degree $n$ and order $n (n - 1) (n - 2)$ , 435
TYPE OF A GROUP, (See SIMPLY ISOMORPHIC GROUPS.)
TYPES OF GROUP, distinct, whose order is
     $24$ , 437–438
     $60$ , 439–440
     $8$ or $16$ , 441, 442
     $p^{2}$ , $p^{3}$ , or $p^{4}$ , where $p$ is an odd prime, 443, 444
     $p^{2} q$ , where $p$ and $q$ are diﬀerent primes, 445–446
     $p q$ , where $p$ and $q$ are diﬀerent primes, 447, 448

WEBER, quoted, 449

YOUNG, quoted, 450, 451, 452, 453