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12 2 e 2.1 2.1.1 S,T S s S T t T (map α α : S T s t = α(s (2.1 S (domain T (codomain (target set, {α(s} T (range (image 2.1.2 s, s S t T s S t T, α s, s S s s, α(s α(s (2.2 α (injection 4 T t T (coimage α t t = α(s S (surjection (bijection S T (cardinal number S T 2 2.1.3 ( Isomorphism, Homomorphism, Automorphism, Endomorphism 4 one to one

13 Figure 2: G H α : G H α(g g = α(g H α(g (2.3 (Group Homomorphism H H 1 G G,1 H H α(1 G = 1 H (2.4 g G α(g 1 = α(g 1 (2.5 ( group isomorphism SO(3. 5 (Automorphism (Endomorphism 5

14 2.1.4 G g G i g i G g 1 = g 1 g. g α g α g : G G : g i α g (g i = g i g (2.6 G ( r G g G G = Gg = (g 1 g, g 2 g g r g (2.7,G G G G G G g i g = g k g ig = g k g i = g i(g 1 q.e.d. α g (g 1 α g (g 2 = g 1 gg 2 g α g (g 1 g 2 (2.8 a α α : g aga 1 (2.9 α(gg = agg a 1 = aga 1 aga 1 = α(gα(g (2.10 α(g = α(g g = g (2.11 1. C 2, C 3 ( 2.

15 1. 2. gg = g\g e a b c e e a b c a a x b b c c (2.12 x x = e g\g e a b c e e a b c gg = a a e c b b b c e c c b e gg = g\g e a b c e e a b c a a e c b b b c a c c b a x = b g\g e a b c e e a b c gg = a a b c e b b c c c e gg = g\g e a b c e e a b c a a b c e b b c e c c e b Z 2 Z 2 (= D 2 x = e x = b x = c Z 4 2.2 G G = {g 1, g 2,, g n }(n G = g 1 + g 2 + + g n = n g i (2.13 1 H = h 1 + h 2 + + h k G( k, Hg g G H Hg = {h 1 g, h 2 g,, h k g} = h 1 g + h 2 g + + h k g (2.14

16 2.2.1 (coset H g 1 = e g i H,(i > 1 G G = r Hg i = Hg 1 + Hg 2 + + Hg r (2.15 i=1 Hg i H ( H\G g i r (index of H G H G (coset decomposition g 1 = e G g 2 G H = Hg 1 g 2 H Hg 2 H Hg 2 G g 3 Hg 3 Hg i {g i } g i+1 H Hg i+1 G Hg i H (i 1 (hg i = h g i = h h 1 g H 1. G = Hg i,g/h = {ah a G} 2. (Lagrange s theorem, G = H G/H (2.16 G (H (G/H 3. e g i k #G = G C k = g i G G/H G /k G 1. C 3v C 3 C 3v = C 3 + C 3 σ 1 2. C 3 σ 1 C 3 σ 2

17 2.2.2 conjugacy class g G a G g(a g 1, g 2 G g G g 1 g 2 H 1, H 2 G g G H 1 H 2 g(a = gag 1 (2.17 g 1 = gg 2 g 1 (2.18 g 1 g 2 (2.19 H 1 = gh 2 g 1 (2.20 H 1 H 2 (2.21 G a [a] = {g 1 ag1 1, g 2 ag2 1,, g n agn 1 } (2.22 a 1. 2. 3. 4. 6 g : a = gbg 1 a b (2.23 5. G/ 6 (a a a (b a b b c a c (c a b b a

18 ( C 3v C 3v e,{c 3, c 1 3 },{σ i } C 3v : C 3v a b bab 1 = C 3v b\a e c 3 c 1 3 σ 1 σ 2 σ 3 e e c 3 c 1 3 σ 1 σ 2 σ 3 c 3 e c 3 c 1 3 σ 3 σ 1 σ 2 c 1 3 e c 3 c 1 3 σ 2 σ 3 σ 1 σ 1 e c 1 3 c 3 σ 1 σ 3 σ 2 σ 2 e c 1 3 c 3 σ 3 σ 2 σ 1 σ 3 e c 1 3 c 3 σ 2 σ 1 σ 3 (2.24 C 1 C 2 C 3 e c 3, c3 1 σ 1, σ 2, σ 3 [e] [c 3 ] [σ 1 ] (2.25 2.2.3 1. C i = g C i g C 3v C 2 = c 3 + c 1 3, C 3 = σ 1 + σ 2 + σ 3. 2. gc i g 1 = C i C 3v σ 1 C 2 σ 1 = σ 1 c 3 σ 1 + σ 1 c 1 3 σ 1. 3. G = C i 4. g G gc i = C i g 5. a, b ab b(abb 1 = ba 6. C i C j = C k ijc k (2.26 2.3

19 2.3.1 H = ghg 1 (2.27 g G ghg 1 = H H g : gh = Hg (2.28 g 1 Hg 2 H = g 1 g 2 HH = g 1 g 2 H H G/H(coset group a a (quotient group 2.3.2 G G ϕ 1. ϕ : G G ϕ G ϕ(g G Im(ϕ G 2. ϕ : G G e G G (Kernel Ker(ϕ G ϕ K = Ker(ϕ G K = Ker(ϕ k K ϕ(gkg 1 = ϕ(gϕ(kϕ(g 1 = e (2.29 gkg 1 K. g G GKG 1 = K K (2.30 K

20 Theorem : ϕ : G G Im(ϕ G/Ker(ϕ f : G/K Im(ϕ G : f(kg ϕ(g. ϕ f(kg 1 Kg 2 = ϕ(g 1 g 2 = ϕ(g 1 ϕ(g 2 = f(kg 1 f(kg 2 (2.31 f Kg 1,Kg 2 f f(kg 1 = f(kg 2 ϕ(g 1 = ϕ(g 2 g 1 g 1 2 = k K g 1 = kg 2 (2.32 Kg 1 = Kkg 2 = Kg 2 g 1, g 2 f 1. C 3 = (e, c 3, c 1 3 C 3v 2. α : C 3v C 2 {e, τ} α : e, c 3, c 1 3 e (2.33 α : σ i τ (2.34 (e, c 3 c 1 3 e C 2 σ 1, σ 2, σ 3 τ α 3. C 3v, C 3 C 3v = C 3 (e, c 3, c 1 3 + C 3 σ 1 ({σ i } (2.35 C 3v /C 3 C 2 1. 2. (2.3 3. C 2 C 3 C 3 σ 1 C 3 C 3 C 3 σ 1 (2.36 C 3 σ 1 C 3 σ 1 C 3

21 2.4 l 2.4.1 n 1 n i p i ( 1 2 n (2.37 p 1 p 2 p n ( ( ( 1 2 3 1 2 3 1 2 3 = 2 3 1 2 1 3 3 2 1 ( 1 2 n q 1 q 2 q n ( 1 2 n p 1 p 2 p n 1. ( p q 2. e = ( i i 3. π = ( i p = = ( i p π 1 = ( p1 p 2 p n p q1 p q2 p qn ( p q1 p q2 p qn 1 2 n ( p i = ( i q ( 1 2 n p 1 p 2 p n (2.38 (2.39 (2.40 4. ( q ( r ( p q ( i p = ( q r ( p ( q ( i p (2.41 n n S n C 3 v ( ( 1 2 3 1 2 3 e = c 1 2 3 3 = ( ( 2 3 1 1 2 3 1 2 3 σ 1 = σ 1 3 2 2 = 3 2 1 C 3 v S 3 ( 1 2 3 c 1 3 = 3 1 2 σ 3 == ( 1 2 3 2 1 3 (2.42

22 2.4.2 1. :(1, 2, 5 = ( 1 2 5 2 5 1 2. ( 1 2 3 4 5 2 5 4 3 1 3. (2, 4 Theorem : ( 1 2 5 (1, 2, 5 = (1, 2(2, 5 = 2 1 5 ( 1 2 3 4 5 2 5 4 3 1 = (1, 2, 5(3, 4 (2.43 ( 1 2 5 1 5 2 (2.44 = (1, 2, 5(3, 4 = (1, 2(2, 5(3, 4 (2.45 4. 5. S n A n Theorem : 2.4.3 Cayley S n = A n + (1, 2A n (2.46 Theorem Cayley : G g 1,...g n g G gg 1,...gg n G g ( g1 g π g = n (2.47 gg 1 gg n π a π b = π ab 2.4.4 1. (1, 5(1, 2, 3(1, 5 = (5, 2, 3 (2.48

23 2. S n n (partition n λ i [λ 1,, λ k ] i λ i = n S 5 [2, 3] [(1, 2(3, 4, 5] (2.49 3. conjugation S n p(n n

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