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Finite Group

A Group of finite Order. Examples of finite groups are the Modulo Multiplication Groups and the Point Groups. The Classification Theorem of finite Simple Groups states that the finite Simple Groups can be classified completely into one of five types.


There is no known Formula to give the number of possible finite groups as a function of the Order $h$. It is possible, however, to determine the number of Abelian Groups using the Kronecker Decomposition Theorem, and there is at least one Abelian Group for every finite order $h$.


The following table gives the numbers and names of the first few groups of Order $h$. In the table, ${\it NA}$ denotes the number of non-Abelian groups, $A$ denotes the number of Abelian Groups, and $N$ the total number of groups. In addition, $Z_n$ denotes an Cyclic Group of Order $n$, $A_n$ an Alternating Group, $D_n$ a Dihedral Group, $Q_8$ the group of the Quaternions, $T$ the cubic group, and $\otimes$ a Direct Product.


$h$ Name $A$ NA $N$
1 Finite Group e 1 0 1
2 Finite Group Z2 1 0 1
3 Finite Group Z3 1 0 1
4 Finite Group Z2Z2, Finite Group Z4 2 0 2
5 Finite Group Z5 1 0 1
6 Finite Group Z6, Finite Group D3 1 1 2
7 Finite Group Z7 1 0 1
8 Finite Group Z2Z2Z2, Finite Group Z2Z4, Finite Group Z8, Finite Group Q8, Finite Group D4 3 2 5
9 $Z_3\otimes Z_3, Z_9$ 2 0 2
10 $Z_{10}, D_5$ 1 1 2
11 $Z_{11}$ 1 0 1
12 $Z_2\otimes Z_6, Z_{12}, A_4, D_6, T$ 2 3 5
13 $Z_{13}$ 1 0 1
14 $Z_{14}, D_7$ 1 1 2
15 $Z_{15}$ 1 0 1


Miller (1930) gave the number of groups for orders 1-100, including an erroneous 297 as the number of groups of Order 64. Senior and Lunn (1934, 1935) subsequently completed the list up to 215, but omitted 128 and 192. The number of groups of Order 64 was corrected in Hall and Senior (1964). James et al. (1990) found 2328 groups in 115 Isoclinism families of Order 128, correcting previous work, and O'Brien (1991) found the number of groups of Order 256. The number of groups is known for orders up to 1000, with the possible exception of 512 and 768. Besche and Eick (1998) have determined the number of finite groups of orders less than 1000 which are not powers of 2 or 3. These numbers appear in the Magma ${}^{\scriptstyle\circledRsymbol}$ database. The numbers of nonisomorphic finite groups $N$ of each Order $h$ for the first few hundred orders are given in the following table (Sloane's A000001--the very first sequence).


The smallest order for which there exist $n=1$, 2, ...nonisomorphic groups are 1, 4, 75, 28, 8, 42, ... (Sloane's A046057). The incrementally largest number of nonisomorphic finite groups are 1, 2, 5, 14, 15, 51, 52, 267, 2328, ... (Sloane's A046058), which occur for orders 1, 4, 8, 16, 24, 32, 48, 64, 128, ... (Sloane's A046059).


The number $A$ of Abelian Groups of Order $h=1$, 2, ... are given by 1, 1, 1, 2, 1, 1, 1, 3, ... (Sloane's A000688). The following table summarizes the total number of finite groups $N$ and the number of Abelian finite groups $A$ for small orders $h$.

$h$ $N$ $A$ $h$ $N$ $A$ $h$ $N$ $A$ $h$ $N$ $A$
1 1 1 51 1 1 101 1 1 151 1 1
2 1 1 52 5 2 102 4 1 152 12 3
3 1 1 53 1 1 103 1 1 153 2 2
4 2 2 54 15 3 104 14 3 154 4 1
5 1 1 55 2 1 105 2 1 155 2 1
6 2 1 56 13 3 106 2 1 156 18 2
7 1 1 57 2 1 107 1 1 157 1 1
8 5 3 58 2 1 108 45 6 158 2 1
9 2 2 59 1 1 109 1 1 159 1 1
10 2 1 60 13 2 110 6 1 160 238 7
11 1 1 61 1 1 111 2 1 161 1 1
12 5 2 62 2 1 112 43 5 162 55 5
13 1 1 63 4 2 113 1 1 163 1 1
14 2 1 64 267 11 114 6 1 164 5 2
15 1 1 65 1 1 115 1 1 165 2 1
16 14 5 66 4 1 116 5 2 166 2 1
17 1 1 67 1 1 117 4 2 167 1 1
18 5 2 68 5 2 118 2 1 168 57 3
19 1 1 69 1 1 119 1 1 169 2 2
20 5 2 70 4 1 120 47 3 170 4 1
21 2 1 71 1 1 121 2 2 171 5 2
22 2 1 72 50 6 122 2 1 172 4 2
23 1 1 73 1 1 123 1 1 173 1 1
24 15 3 74 2 1 124 4 2 174 4 1
25 2 2 75 3 2 125 5 3 175 2 2
26 2 1 76 4 2 126 16 2 176 42 5
27 5 3 77 1 1 127 1 1 177 1 1
28 4 2 78 6 1 128 2328 15 178 2 1
29 1 1 79 1 1 129 2 1 179 1 1
30 4 1 80 52 5 130 4 1 180 37 4
31 1 1 81 15 5 131 1 1 181 1 1
32 51 7 82 2 1 132 10 2 182 4 1
33 1 1 83 1 1 133 1 1 183 2 1
34 2 1 84 15 2 134 2 1 184 12 3
35 1 1 85 1 1 135 5 3 185 1 1
36 14 4 86 2 1 136 15 3 186 6 1
37 1 1 87 1 1 137 1 1 187 1 1
38 2 1 88 12 3 138 4 1 188 4 2
39 2 1 89 1 1 139 1 1 189 13 3
40 14 3 90 10 2 140 11 2 190 4 1
41 1 1 91 1 1 141 1 1 191 1 1
42 6 1 92 4 2 142 2 1 192 1543 11
43 1 1 93 2 1 143 1 1 193 1 1
44 4 2 94 2 1 144 197 1 194 2 1
45 2 2 95 1 1 145 1 1 195 2 1
46 2 1 96 230 7 146 2 1 196 17 4
47 1 1 97 1 1 147 6 2 197 1 1
48 52 5 98 5 2 148 5 2 198 10 2
49 2 2 99 2 2 149 1 1 199 1 1
50 2 2 100 16 4 150 13 2 200 52 6

$h$ $N$ $A$ $h$ $N$ $A$ $h$ $N$ $A$ $h$ $N$ $A$
201 2 1 251 1 1 301 2 1 351 14 3
202 2 1 252 46 4 302 2 1 352 195 7
203 2 1 253 2 1 303 1 1 353 1 1
204 12 2 254 2 1 304 42 5 354 4 1
205 2 1 255 1 1 305 2 1 355 2 1
206 2 1 256 56092 22 306 10 2 356 5 2
207 2 2 257 1 1 307 1 1 357 2 1
208 51 5 258 6 1 308 9 2 358 2 1
209 1 1 259 1 1 309 2 1 359 1 1
210 12 1 260 15 2 310 6 1 360 162 6
211 1 1 261 2 2 311 1 1 361 2 2
212 5 2 262 2 1 312 61 3 362 2 1
213 1 1 263 1 1 313 1 1 363 3 2
214 2 1 264 39 3 314 2 1 364 11 2
215 1 1 265 1 1 315 4 2 365 1 1
216 177 9 266 4 1 316 4 2 366 6 1
217 1 1 267 1 1 317 1 1 367 1 1
218 2 1 268 4 2 318 4 1 368 42 5
219 2 1 269 1 1 319 1 1 369 2 2
220 15 2 270 30 3 320 1640 11 370 4 1
221 1 1 271 1 1 321 1 1 371 1 1
222 6 1 272 54 5 322 4 1 372 15 2
223 1 1 273 5 1 323 1 1 373 1 1
224 197 7 274 2 1 324 176 10 374 4 1
225 6 4 275 4 2 325 2 2 375 7 3
226 2 1 276 10 2 326 2 1 376 12 3
227 1 1 277 1 1 327 2 1 377 1 1
228 15 2 278 2 1 328 15 3 378 60 3
229 1 1 279 4 2 329 1 1 379 1 1
230 4 1 280 40 3 330 12 1 380 11 2
231 2 1 281 1 1 331 1 1 381 2 1
232 14 3 282 4 1 332 4 2 382 2 1
233 1 1 283 1 1 333 5 2 383 1 1
234 16 2 284 4 2 334 2 1 384 20169 15
235 1 1 285 2 1 335 1 1 385 2 1
236 4 2 286 4 1 336 228 5 386 2 1
237 2 1 287 1 1 337 1 1 387 4 2
238 4 1 288 1045 14 338 5 2 388 5 2
239 1 1 289 2 2 339 1 1 389 1 1
240 208 5 290 4 1 340 15 2 390 12 1
241 1 1 291 2 1 341 1 1 391 1 1
242 5 2 292 5 2 342 18 2 392 44 6
243 67 7 293 1 1 343 5 3 393 1 1
244 5 2 294 23 2 344 12 3 394 2 1
245 2 2 295 1 1 345 1 1 395 1 1
246 4 1 296 14 3 346 2 1 396 30 4
247 1 1 297 5 3 347 1 1 397 1 1
248 12 3 298 2 1 348 12 2 398 2 1
249 1 1 299 1 1 349 1 1 399 5 1
250 15 3 300 49 4 350 10 2 400 221 10

See also Abelian Group, Abel's Theorem, Abhyankar's Conjecture, Alternating Group, Burnside's Lemma, Burnside Problem, Chevalley Groups, Classification Theorem, Composition Series, Dihedral Group, Group, Jordan-Hölder Theorem, Kronecker Decomposition Theorem, Lie Group, Lie-Type Group, Linear Group, Modulo Multiplication Group, Order (Group), Orthogonal Group, p-Group, Point Groups, Simple Group, Sporadic Group, Symmetric Group, Symplectic Group, Twisted Chevalley Groups, Unitary Group


References

Arfken, G. ``Discrete Groups.'' §4.9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 243-251, 1985.

Artin, E. ``The Order of the Classical Simple Groups.'' Comm. Pure Appl. Math. 8, 455-472, 1955.

Aschbacher, M. Finite Group Theory. Cambridge, England: Cambridge University Press, 1994.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 73-75, 1987.

Besche and Eick. ``Construction of Finite Groups.'' To Appear in J. Symb. Comput.

Besche and Eick. ``The Groups of Order at Most 1000.'' To Appear in J. Symb. Comput.

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, 1985.

Hall, M. Jr. and Senior, J. K. The Groups of Order $2^n (n\leq 6)$. New York: Macmillan, 1964.

James, R.; Newman, M. F.; and O'Brien, E. A. ``The Groups of Order 128.'' J. Algebra 129, 136-158, 1990.

Miller, G. A. ``Determination of All the Groups of Order 64.'' Amer. J. Math. 52, 617-634, 1930.

O'Brien, E. A. ``The Groups of Order 256.'' J. Algebra 143, 219-235, 1991.

O'Brien, E. A. and Short, M. W. ``Bibliography on Classification of Finite Groups.'' Manuscript, Australian National University, 1988.

Senior, J. K. and Lunn, A. C. ``Determination of the Groups of Orders 101-161, Omitting Order 128.'' Amer. J. Math. 56, 328-338, 1934.

Senior, J. K. and Lunn, A. C. ``Determination of the Groups of Orders 162-215, Omitting Order 192.'' Amer. J. Math. 57, 254-260, 1935.

Simon, B. Representations of Finite and Compact Groups. Providence, RI: Amer. Math. Soc., 1996.

Sloane, N. J. A. Sequences A000001/M0098, A000688/M0064, A046057, A046058, and A046059in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

University of Sydney Computational Algebra Group. ``The Magma Computational Algebra for Algebra, Number Theory and Geometry.'' http://www.maths.usyd.edu.au:8000/u/magma/.

mathematica.gif Weisstein, E. W. ``Groups.'' Mathematica notebook Groups.m.

Wilson, R. A. ``ATLAS of Finite Group Representation.'' http://for.mat.bham.ac.uk/atlas/.


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© 1996-9 Eric W. Weisstein
1999-05-26