“Finite Group”的概念、定义、习题解题方法及翻译-数学辞典

A Group of finite Order. Examples of finite groups are the Modulo MultiplicationGroups and the Point Groups. The Classification Theorem of finiteSimple Groups states that the finite Simple Groups can beclassified 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

. It is possible, however, to determine the number of Abelian Groups using theKronecker Decomposition Theorem, and there is at least one Abelian Group for every finite order

The following table gives the numbers and names of the first few groups of Order

. In the table,

denotes the number of non-Abelian groups,

denotes the number of Abelian Groups, and

the total number of groups. In addition,

denotes an Cyclic Group of Order

an Alternating Group,

a Dihedral Group,

the group of the Quaternions,

the cubic group, and

a Direct Product.

	Name		NA
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		2	0	2
10		1	1	2
11		1	0	1
12		2	3	5
13		1	0	1
14		1	1	2
15		1	0	1

Miller (1930) gave the number of groups for orders 1-100, including an erroneous 297 as the number of groups ofOrder 64. Senior and Lunn (1934, 1935) subsequently completed the list up to 215, but omitted 128and 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 ofgroups is known for orders up to 1000, with the possible exception of 512 and 768. Besche and Eick (1998) have determinedthe number of finite groups of orders less than 1000 which are not powers of 2 or 3. These numbers appear in the Magma

database. The numbers of nonisomorphic finite groups

of each Order

forthe first few hundred orders are given in the following table (Sloane's A000001--the very first sequence).

The smallest order for which there exist

, 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

of Abelian Groups of Order

, 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

and thenumber of Abelian finite groups

for small orders


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


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

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