위수가 30 이하인 유한군의 분류

[폭풍속으로]

Groups of small order Compiled by John Pedersen, Dept of Mathematics, University of South Florida, jfp@math.usf.edu Order 1 and all prime orders (1 group: 1 abelian, 0 nonabelian) All groups of prime order p are isomorphic to C_p, the cyclic group of order p. A concrete realization of this group is Z_p, the integers under addition modulo p. Order 4 (2 groups: 2 abelian, 0 nonabelian) C_4, the cyclic group of order 4 V = C_2 x C_2 (the Klein four group) = symmetries of a rectangle. A presentation for the group is <a, b; a^2 = b^2 = (ab)^2 = 1> The Cayley table of the group is (putting c = ab): | 1 a b c --+----------- 1 | 1 a b c a | a 1 c b b | b c 1 a c | c b a 1 A matrix representation is the four 2x2 matrices [1 0] [1 0] [-1 0] [-1 0] [0 1], [0 -1], [ 0 1], [ 0 -1] A permutation representation is the following four elements of S_4: (1), (1 2)(3 4), (1 3)(2 4) and (1 4)(2 3). Its lattice of subgroups is (in the notation of the Cayley table) V / | \ <a> <b> <c> \ | / {1} Order 6 (2 groups: 1 abelian, 1 nonabelian) C_6 S_3, the symmetric group of degree 3 = all permutations on three objects, under composition. In cycle notation for permutations, its elements are (1), (1 2), (1 3), (2, 3), (1 2 3) and (1 3 2). There are four proper subgroups of S_3; they are all cyclic. There are the three of order 2 generated by (1 2), (1 3) and (2 3), and the one of order 3 generated by (1 2 3). Only the one of order 3 is normal in S_3. A presentation for S_3 is (where s corresponds to (1 2) and t to (2 3)): <s,t; s^2 = t^2 = 1, sts = tst> Another presentation (with s <-> (1 2 3), t <-> (1 2)) is <s,t; s^3 = t^2 = 1, ts = s^2 t> In terms of this second presentation, with 2 = s^2, u = ts and v = ts^2, the Cayley table is | 1 s 2 t u v --+----------------------- 1 | 1 s 2 t u v s | s 2 1 v t u 2 | 2 1 s u v t t | t u v 1 s 2 u | u v t 2 1 s v | v t u s 2 1 This shows S_3 is isomorphic to D_3, the dihedral group of degree 3, that is, the symmetries of an equilateral triangle (this never happens for n > 3). The lattice of subgroups of S_3 is S_3 / / | \ <t> <u> <v> <s> \ \ | / {1} The first three proper subgroups have order two, while <s> has order three and is the only normal one. The center of S_3 is trivial (in fact Z(S_n) is trivial for all n.) The automorphism group of S_3 is isomorphic to S_3. Order 8 (5 groups: 3 abelian, 2 nonabelian) C_8 C_4 x C_2 C_2 x C_2 x C_2 D_4, the dihedral group of degree 4, or octic group. It has a presentation <s, t; s^4 = t^2 = e; ts = s^3 t> In terms of these generators (s corresponds to rotation by pi/2 and t to a reflection about an axis through a vertex), the eight elements are 1,s,s^2,s^3,t,ts,ts^2 and ts^3. Using the notation 2 = s^2, 3 = s^3, t2 = ts^2 and t3 = ts^3, the Cayley table is | 1 s 2 3 t ts t2 t3 --+------------------------ 1 | 1 s 2 3 t ts t2 t3 s | s 2 3 1 t3 t ts t2 2 | 2 3 1 s t2 t3 t ts 3 | 3 1 s 2 ts t2 t3 t t | t ts t2 t3 1 s 2 3 ts |ts t2 t3 t 3 1 s 2 t2 |t2 t3 t ts 2 3 1 s t3 |t3 t ts t2 s 2 3 1 Its subgroup lattice is D_4 / | \ {1,s^2,t,ts^2} <s> {1,s^2,st,ts} / | \ | / | \ <ts^2> <t> <s^2> <st> <ts> \ \ | / / {1} Of these, the proper normal subgroups are the three of order four and <s^2> of order two. The center of D_4 is {1,s^2}, which is also its derived group. The automorphism group of D_4 is isomorphic to D_4. Q, the quaternion group. It has a presentation <s, t; s^4 = 1, s^2 = t^2, sts = t> Q can be realized as consisting of the eight quaternions 1, -1, i, -i, j, -j, k, -k, where i is the imaginary square root of -1, and j and k also obey j^2 = k^2 = -1. These quaternions multiply according to clockwise movement around the figure i / \ k ---- j For example, ij = k and ji = -k (negative because anticlockwise). A matrix representation is given by s and t in the above presentation corresponding to these two 2x2 matrices over the complex numbers: s = [i 0] t = [0 i] [0 -i] [i 0] The subgroup lattice of Q is Q / | \ <s> <st> <t> \ | / <s^2> | {1} All of these subgroups are normal in Q. The center of Q is {1,s^2}, which is also its derived group. The automorphism group of Q is isomorphic to S_4. Order 9 (2 groups: 2 abelian, 0 nonabelian) C_9 C_3 x C_3 Order 10 (2 groups: 1 abelian, 1 nonabelian) C_10 D_5 Order 12 (5 groups: 2 abelian, 3 nonabelian) C_12 C_6 x C_2 A_4, the alternating group of degree 4, consisting of the even permutations in S_4. The subgroup lattice of A_4 is A_4 / \ \ \ \ <(12)(34),(13)(24)> <(123)> <(124)> <(134)> <(234)> / | \ | / / / <(12)(34)> <(13)(24)> <(14)(23)> | / / / \ \ \ / / / / {1} The only proper normal subgroup is <(12)(34),(13)(24)>. D_6, isomorphic to S_3 x C_2 = D_3 x C_2 T which has the presentation <s, t; s^6 = 1, s^3 = t^2, sts = t> T is the semidirect product of C_3 by C_4 by the map g : C_4 -> Aut(C_3) given by g(k) = a^k, where a is the automorphism a(x) = -x. Another presentation for T is <x,y; x^4 = y^3 = 1, yxy = x> In terms of these generators, using AB for x^A y^B, the Cayley table for T is | 00 10 20 30 01 02 11 21 31 12 22 32 ------+----------------------------------------------- 1 = 00| 00 10 20 30 01 02 11 21 31 12 22 32 x = 10| 10 20 30 00 11 12 21 31 01 22 32 02 x^2 = 20| 20 30 00 10 21 22 31 01 11 32 02 12 x^3 = 30| 30 00 10 20 31 32 01 11 21 02 12 22 y = 01| 01 12 21 32 02 00 10 22 30 11 20 31 y^2 = 02| 02 11 22 31 00 01 12 20 32 10 21 30 xy = 11| 11 22 31 02 12 10 20 32 00 21 30 01 x^2y = 21| 21 32 01 12 22 20 30 02 10 31 00 11 x^3y = 31| 31 02 11 22 32 30 00 12 20 01 10 21 xy^2 = 12| 12 21 32 01 10 11 22 30 02 20 31 00 x^2y^2 = 22| 22 31 02 11 20 21 32 00 12 30 01 10 x^3y^2 = 32| 32 01 12 21 30 31 02 10 22 00 11 20 A 2x2 matrix representation of this group over the complex numbers is given by [0 i] [w 0 ] x <--> [i 0] y <--> [0 w^2] where i is a square root of -1 and w is nonreal cube root of 1, for example w = e^{2\pi i/3}. Order 14 (2 groups: 1 abelian, 1 nonabelian) C_14 D_7 Order 15 (1 group: 1 abelian, 0 nonabelian) C_15. Order 16 (14 groups: 5 abelian, 9 nonabelian) C_16 C_8 x C_2 C_4 x C_4 C_4 x C_2 x C_2 C_2 x C_2 x C_2 x C_2 D_8 D_4 x C_2 Q x C_2, where Q is the quaternion group The quasihedral (or semihedral) group of order 16, with presentation <s,t; s^8 = t^2 = 1, st = ts^3> The modular group of order 16, with presentation <s,t; s^8 = t^2 = 1, st = ts^5> The elements are s^k t^m, k = 0,1,...,7, m = 0,1. The center is {1,s^2,s^4,s^6}. Its subgroup lattice is G / | \ <s^2,t> <s> <st> / | \ | / <s^4,t> <s^2t> <s^2> / | \ | / <t> <s^4t> <s^4> \ | / {1} This is the same subgroup lattice structure as for the lattice of subgroups of C_8 x C_2, although the groups are of course nonisomorphic. The automorphism group is isomorphic to D_4 x C_2 Reference: Weinstein, Examples of Groups, pp. 120-123. The group with presentation < s,t; s^4 = t^4 = 1, st = ts^3 > The elements are s^i t^j for i,j = 0,1,2,3. The center of G is {1,s^2,t^2,s^2t^2}. Reference: Weinstein, pp. 124--128. The group with presentation <a,b,c; a^4 = b^2 = c^2 = 1, cbca^2b = 1, bab = a, cac = a> The group G_{4,4} with presentation <s,t; s^4 = t^4 = 1, stst = 1, ts^3 = st^3 > The generalized quaternion group of order 16 with presentation <s,t; s^8 = 1, s^4 = t^2, sts = t > Order 18 (5 groups: 2 abelian, 3 nonabelian) C_18 C_6 x C_3 D_9 S_3 x C_3 The semidirect product of C_3 x C_3 with C_2 which has the presentation <x,y,z; x^2 = y^3 = z^3 = 1, yz = zy, yxy = x, zxz = x> Order 20 (5 groups: 2 abelian, 3 nonabelian) C_20 C_10 x C_2 D_10 The semidirect product of C_5 by C_4 which has the presentation <s,t; s^4 = t^5 = 1, tst = s> The Frobenius group of order 20, with presentation <s,t; s^4 = t^5 = 1, ts = st^2> This is the Galois group of x^5 -2 over the rationals, and can be represented as the subgroup of S_5 generated by (2 3 5 4) and (1 2 3 4 5). Order 21 (2 groups: 1 abelian, 1 nonabelian) C_21 <a,b; a^3 = b^7 = 1, ba = ab^2> This is the Frobenius group of order 21, which can be represented as the subgroup of S_7 generated by (2 3 5)(4 7 6) and (1 2 3 4 5 6 7), and is the Galois group of x^7 - 14x^5 + 56x^3 -56x + 22 over the rationals (ref: Dummit & Foote, p.557). Order 22 (2 groups: 1 abelian, 1 nonabelian) C_22 D_11 Order 24 (15 groups: 3 abelian, 12 nonabelian) C_24 C_2 x C_12 C_2 x C_2 x C_6 S_4 S_3 x C_4 S_3 x C_2 x C_2 D_4 x C_3 Q x C_3 A_4 x C_2 T x C_2 Five more nonabelian groups of order 24 Reference: Burnside, pp. 157--161. Order 25 (2 groups: 2 abelian, 0 nonabelian) C_25 C_5 x C_5 Order 26 (2 groups: 1 abelian, 1 nonabelian) C_26 D_13 Order 27 (5 groups: 3 abelian, 2 nonabelian) C_27 C_9 x C_3 C_3 x C_3 x C_3 The group with presentation <s,t; s^9 = t^3 = 1, st = ts^4 > The group with presentation <x,y,z; x^3 = y^3 = z^3 = 1, yz = zyx, xy = yx, xz = zx> Reference: Burnside, p. 145. Order 28 (4 groups: 2 abelian, 2 nonabelian) C_28 C_2 x C_14 D_14 D_7 x C_2 Order 30 (4 groups: 1 abelian, 3 nonabelian) C_30 D_15 D_5 x C_3 D_3 x C_5 Reference: Dummit & Foote, pp. 183-184. A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu

[체 게바라]

여기서 C_n 이란 위수 n 인 순환군을 가르키는 것이겠군요??? Zn으로 써도 무난할까요???

[폭풍속으로]

맞습니다. C_n 은 위수가 n 인 순환군을 말합니다. 동형에 관계없이 위수가 n 인 부분군은 Z_n 으로 유일하므로 Z_n 으로 써도 무방합니다. 책에 따라서는 Z_n 보다는 C_n 으로 쓰는 경우가 있기도 합니다. 참고로, C_n = { e, x, x^2, x^3, ... , x^(n-1) } 로 나타냅니다. 경문사 현대대수학 교재에 소개되어 있습니다.