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Finite Group

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A finite group is a group having finite group order. Examples of finite groups are the modulo multiplication groups, point groups, cyclic groups, dihedral groups, symmetric groups, alternating groups, and so on.

Properties of finite groups are implemented in the Wolfram Language as FiniteGroupData[group, prop].

The classification theorem of finite groups states that the finite simple groups can be classified completely into one of five types.

FiniteGroups8

A convenient way to visualize groups is using so-called cycle graphs, which show the cycle structure of a given abstract group. For example, cycle graphs of the 5 nonisomorphic groups of order 8 are illustrated above (Shanks 1993, p. 85).

Frucht's theorem states that every finite group is the graph automorphism group of a finite undirected graph.

The finite (cyclic) group C_2 forms the subject for the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern University mathematics department a capella group "The Klein Four."

The following table gives the numbers and names of the distinct groups of group order h for small h. In the table, C_n denotes an cyclic group of group order n, × a group direct product, D_n a dihedral group, Q_8 the quaternion group, A_n an alternating group, T the non-Abelian finite group of order 12 that is not A_4 and not D_6 (and is not the purely rotational subgroup T of the point group T_h), G_(16)^((4)) the quasihedral (or semihedral) group of order 16 with group presentation <s,t;s^8=t^2=1,st=ts^3>, G_(16)^((5)) the modular group of order 16 with group presentation <s,t;s^8=t^2=1,st=ts^5>, G_(16)^((6)) the group of order 16 with group presentation <s,t;s^4=t^4=1,st=ts^3>, G_(16)^((7)) the group of order 16 with group presentation <a,b,c;a^4=b^2=c^2=1,cbca^2b=1,bab=a,cac=a>, G_(16)^((8)) the group G_(4,4) with group presentation <s,t;s^4=t^4=1,stst=1,ts^3=st^3>, G_(16)^((9)) the generalized quaternion group of order 16 with group presentation <s,t;s^8=1,s^4=t^2,sts=t>, S_n a symmetric group, G_(18)^((3)) the semidirect product of C_3×C_3 with C_2 with group presentation <x,y,z;x^2=y^3=z^3=1,yz=zy,yxy=x,zxz=x>, F_n the Frobenius group of order n, G_(20)^((3)) the semidirect product of C_5 by C_4 with group presentation <s,t;s^4=t^5=1,tst=s>, G_(27)^((1)) the group with group presentation <s,t;s^9=t^3=1,st=ts^4>, G_(27)^((2)) the group with group presentation <x,y,z;x^3=y^3=z^3=1,yz=zyx,xy=yx,xz=zx>, and G_(28)^((2)) the semidirect product of C_7 by C_4 with group presentation <s,t;s^4=t^7=1,tst=s>

h#Abelian#non-Abeliantotal
11<e>0-1
21C_20-1
31C_30-1
42C_4, C_2×C_20-2
51C_50-1
61C_61D_32
71C_70-1
83C_8, C_2×C_4, C_2×C_2×C_22D_4, Q_85
92C_9, C_3×C_30-2
101C_(10)1D_52
111C_(11)0-1
122C_(12), C_2×C_63A_4, D_6, T5
131C_(13)0-1
141C_(14)1D_72
151C_(15)0-1
165C_(16), C_8×C_2, C_4×C_4, C_4×C_2×C_2, C_2×C_2×C_2×C_29D_8, D_4×C_2, Q×C_2, G_(16)^((4)), G_(16)^((5)), G_(16)^((6)), G_(16)^((7)), G_(16)^((8)), G_(16)^((9))14
171C_(17)0-1
182C_(18), C_6×C_33D_9, S_3×C_3, G_(18)^((3))5
191C_(19)0-1
202C_(20), C_(10)×C_23D_(10), F_(20), G_(20)^((2))5
211C_(21)1F_(21)2
221C_(22)1D_(11)2
231C_(23)0-1
243C_(24), C_2×C_(12), C_2×C_2×C_612S_4, S_3×C_4, S_3×C_2×C_2, D_4×C_3, Q×C_3, A_4×C_2, T×C_2, plus 5 others15
252C_(25), C_5×C_50-2
261C_(26)1D_(13)2
273C_(27), C_9×C_3, C_3×C_3×C_32G_(27)^((1)), G_(27)^((2))5
282C_(28), C_2×C_(14)2D_(14), G_(28)^((2))4
291C_(29)0-1
304C_(30)3D_(15), D_5×C_3, D_3×C_54
311C_(31)0-1

The following table lists some properties of small finite groups. Here h is again the group order, PG indicates that a group can be generated by a single permutation, MMG indicates that a group is a modulo multiplication group, C is the number of conjugacy classes, S is the number of subgroups, and N is the number of normal subgroups. Note that the smallest groups that are neither permutation nor modulo multiplication groups are Q_8, C_3×C_3, and T.

hgroupAbelianPGMMGCC lengthsSS lengthsNcounts of A s.t. A^i=1
1<e>yesyesyes111111
2C_2yesyesno22×121, 221, 2
3C_3yesyesyes33×121, 321, 1, 3
4C_4yesyesyes44×131, 2, 431, 2, 1, 4
C_2×C_2yesnoyes44×151, 3×2, 451, 4, 1, 4
5C_5yesyesno55×121, 421, 1, 1, 1, 5
6C_6yesyesyes66×141, 2, 3, 641, 2, 3, 2, 1, 6
D_3noyesno31, 2, 361, 3×2, 3, 631, 4, 3, 4, 1, 6
7C_7yesyesno77×121, 721, 1, 1, 1, 1, 1, 7
8C_8yesyesyes88×141, 2, 4, 841, 2, 1, 4, 1, 2, 1, 8
C_2×C_4yesnoyes88×181, 3×2, 3×4, 841, 4, 1, 8, 1, 4, 1, 8
C_2×C_2×C_2yesnoyes88×1161, 7×2, 7×4, 841, 8, 1, 8, 1, 8, 1, 8
D_4noyesno52×1, 3×2101, 5×2, 3×4, 861, 6, 1, 8, 1, 6, 1, 8
Q_8nonono52×1, 3×261, 2, 3×4, 861, 2, 1, 8, 1, 2, 1, 8
9C_9yesyesno99×131, 3, 931, 1, 3, 1, 1, 3, 1, 1, 9
C_3×C_3yesnono
10C_(10)yesyesyes1010×141, 2, 5, 1041, 2, 1, 2, 5, 2, 1, 2, 1, 10
D_5noyesno41, 2×2, 581, 5×2, 5, 1031, 6, 1, 6, 5, 6, 1, 6, 1, 10
11C_(11)yesyesno1111×121, 1121, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11
12C_(12)yesyesyes1212×161, 2, 3, 4, 6, 1261, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12
C_2×C_6yesnoyes1212×1101, 3×2, 3, 4, 3×6, 12101, 4, 3, 4, 1, 12, 1, 4, 3, 4, 1, 12
A_4noyesno41, 3, 2×4101, 3×2, 4×3, 4, 1231, 4, 9, 4, 1, 12, 1, 4, 9, 4, 1, 12
D_6noyesno62×1, 2×2, 2×3161, 7×2, 3, 3×4, 3×6, 1281, 8, 3, 8, 1, 12, 1, 8, 3, 8, 1, 12
Tnonono62×1, 2×2, 2×381, 2, 3, 3×4, 6, 1231, 2, 3, 8, 1, 6, 1, 8, 3, 2, 1, 12
13C_(13)yesyesyes1313×121, 1321, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13
14C_(14)yesyesno1414×141, 2, 7, 1441, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 14
D_7noyesno51, 3×2, 7101, 7×2, 7, 1431, 8, 1, 8, 1, 8, 7, 8, 1, 8, 1, 8, 1, 14
15C_(15)yesyesno1515×141, 3, 5, 1541, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 15

The problem of determining the nonisomorphic finite groups of order h was first considered by Cayley (1854). There is no known formula to give the number of possible finite groups g(h) as a function of the group order h. However, there are simple formulas for special forms of h.

g(1)=1
(1)
g(p)=1
(2)
g(pq)={1 if p(q-1); 2 if p|(q-1)
(3)
g(p^2)=2
(4)
g(p^3)=5,
(5)

where p and q>p are distinct primes. In addition, there is a beautiful algorithm due to Hölder (Hölder 1895, Alonso 1976) for determining g(n) for squarefree n, namely

g(n)=sum_(d|n)product_(p|d; d!=1)(p^(o_p(n/d))-1)/(p-1),
(6)

where o_p(m) is the number of primes q such that q|m and p|(q-1) (Dennis).

Miller (1930) gave the number of groups for orders 1-100, including an erroneous 297 as the number of groups of group order 64. Senior and Lunn (1934, 1935) subsequently completed the list up to 215, but omitted 128 and 192. The number of groups of group order 64 was corrected in Hall and Senior (1964). James et al. (1990) found 2328 groups in 115 isoclinism families of group order 128, correcting previous work, and O'Brien (1991) found the number of groups of group order 256. Currently, the number of groups is known for orders up to 2047, with the difficult cases of orders 512 (g(512)=10494213; Eick and O'Brien 1999b), 768 (Besche and Eick 2001ab), and 1024 now put to rest (Conway et al. 2008). The numbers of nonisomorphic finite groups N of each group order h for the first few hundred orders are given in the table below (OEIS A000001--the very first sequence). The number of nonisomorphic groups of orders 2^n for n=0, 1, ... are 1, 1, 2, 5, 14, 51, 267, 2328, 56092, ... (OEIS A000679).

The smallest orders h for which there exist n=1, 2, ... nonisomorphic groups are 1, 4, 75, 28, 8, 42, ... (OEIS A046057). The incrementally largest number of nonisomorphic finite groups are 1, 2, 5, 14, 15, 51, 52, 267, 2328, ... (OEIS A046058), which occur for orders 1, 4, 8, 16, 24, 32, 48, 64, 128, ... (OEIS A046059). Dennis has conjectured that the number of groups g(h) of order h assumes every positive integer as a value an infinite number of times.

It is simple 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 number A of Abelian groups of group order h=1, 2, ... are given by 1, 1, 1, 2, 1, 1, 1, 3, ... (OEIS A000688). The following table summarizes the total number of finite groups N and the number of Abelian finite groups A for orders h from 1 to 400. A table of orders up to 1000 is given by Royle; the GAP software package includes a table of the number of finite groups up to order 2000, excluding 1024. The number of finite groups of a given order is implemented in the Wolfram Language as FiniteGroupCount[n].

hNAhNAhNAhNA
11151111011115111
211525210241152123
31153111031115322
4225415310414315441
51155211052115521
6215613310621156182
71157211071115711
853582110845615821
92259111091115911
102160132110611602387
111161111112116111
12526221112435162555
131163421131116311
142164267111146116452
151165111151116521
1614566411165216621
171167111174216711
1852685211821168573
191169111191116922
2052704112047317041
212171111212217152
2221725061222117242
231173111231117311
2415374211244217441
252275321255317522
26217642126162176425
275377111271117711
2842786112823281517821
291179111292117911
30418052513041180374
3111811551311118111
32517822113210218241
331183111331118321
34218415213421184123
351185111355318511
36144862113615318661
371187111371118711
3821881231384118842
3921891113911189133
401439010214011219041
411191111411119111
4261924214221192154311
431193211431119311
444294211441971019421
452295111451119521
462196231714621196174
471197111476219711
48525985214852198102
492299221491119911
5052100164150132200526
hNAhNAhNAhNA
201212511130121351143
20221252464302213521957
20321253213031135311
2041222542130442535441
20521255113052135521
20621256560922230610235652
20722257113071135721
208515258613089235821
20911259113092135911
210121260152310613601626
21111261223111136122
212522622131261336221
21311263113131136332
2142126439331421364112
21511265113154236511
2161779266413164236661
21711267113171136711
218212684231841368425
21921269113191136922
22015227030332016401137041
22111271113211137111
2226127254532241372152
22311273513231137311
2241977274213241761037441
22564275423252237573
2262127610232621376123
22711277113272137711
22815227821328153378603
22911279423291137911
23041280403330121380112
23121281113311138121
232143282413324238221
23311283113335238311
23416228442334213842016915
23511285213351138521
2364228641336228538621
23721287113371138742
238412881045143385238852
23911289223391138911
240208529041340152390121
24111291213411139111
2425229252342182392446
243677293113435339311
2445229423234412339421
24522295113451139511
2464129614334621396304
24711297533471139711
2481232982134812239821
24911299113491139951
25015330049435010240022110

See also

Abelian Group, Abhyankar's Conjecture, Alternating Group, Burnside Problem, Cauchy-Frobenius Lemma, Chevalley Groups, Classification Theorem of Finite Groups, Composition Series, Continuous Group, Crystallographic Point Groups, Cycle Graph, Cyclic Group, Dihedral Group, Discrete Group, Feit-Thompson Theorem, Frucht's Theorem, Group, Group Order, Infinite Group, Jordan-Hölder Theorem, Kronecker Decomposition Theorem, Lie Group, Lie-Type Group, Linear Group, Modulo Multiplication Group, Orthogonal Group, p-Group, Point Groups, Quaternion Group, Simple Group, Sporadic Group, Symmetric Group, Symplectic Group, Twisted Chevalley Groups, Unitary Group Explore this topic in the MathWorld classroom

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References

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Finite Group

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Weisstein, Eric W. "Finite Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FiniteGroup.html

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