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1 Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and j to, others untouched) and are of order 2. Three of them are exchanges of two pars of objects (, j)(k, l) ( to j and j to, k to l and l to k), stll of order 2. Eght of them are cyclc permutatons of three objects (, j, k) ( to j, j to k, k to, the other one untouched) and are of order 3. Sx of them are cyclc permutatons of four objects (, j, k, l) ( to j to k to l to ) and are of order 4. Obvously ths s a dfferent group than D 4, but we can dentfy D 4 as a subgroup of S 4 correspondng to the subset of elements {e, (A, C), (B, D), (A, B)(C, D), (A, D)(B, C), (A, C)(B, D), (A, B, C, D), (A, D, C, B)} (1) That s, D 4 can be dentfed as a subgroup of S 4. Cayley s Theorem Every fnte group of order n can be consdered as (s somorphc to) a subgroup of S n. To prove ths theorem, consder the n group elements of group G as the objects that are beng permuted by S n. We need to demonstrate the correspondence between the group elements of G and that of a subgroup of S n. Frst, notce that left multplcaton of an element g G on all group elements {h} of G corresponds to a permutaton of these n objects. Ths s because, frstly, f gh 1 = gh 2, then h 1 = h 2, whch can be obtaned by multplyng g 1 on both sdes of the frst equaton. Ths mples that after left multplcaton of g, there are n dfferent elements n {gh, h G} and they all belong to G. Therefore, left multplcaton corresponds to permutaton of the n group elements n G. Secondly, the permutaton operaton P g obtaned wth left multplcaton of g G forms a group where the composton of P g1 and P g2 s smply the permutaton operaton obtaned wth left multplcaton of g 1 g 2. The dentty operaton s P e. The nverse of P g s P g 1. And t s easy to verfy that the composton of P g s s assocatve. * Note that ths embeddng of a group of order n nto the permutaton group S n s dfferent from the prevous embeddng of D 4 (whch has 8 elements) nto S 4. 1
2 2.4 The group of ntegers Z At the begnnng of the class, we mentoned that the set of all ntegers Z form a group. The composton rule s addton and the dentty element s 0. Ths group s dfferent from all the prevous examples n that there are an nfnte number of elements. The group s stll dscrete but not fnte. The Z group can be thought of as the n lmt of the Z n groups. To take nto account nfnte groups lke Z, the generatng set of a group needs to be more rgorously defned as a subset such that every element of the group can be expressed as the combnaton (under the group operaton) of fntely many elements of the subset and ther nverses. Under ths defnton, we can choose ether {1} or { 1} as the generatng set of the group of ntegers. 2.5 Crcle group The symmetry group of a drected crcle. Fgure 1: Drected crcle A drected crcle has a contnuous rotaton symmetry. The crcle s nvarant under rotaton by any angle θ [0, 2π). The composton of two rotaton operatons corresponds to the addton of two angles θ 1 + θ 2 (mod 2π). If we use the exponental e θ to represent the group element, then the group elements correspond to complex numbers of absolute value 1. The composton rule becomes multplcaton e θ 1 e θ 2 = e (θ 1+θ 2 ). The crcle group has an nfnte number of elements and the elements are contnuous. Therefore, the crcle group s sad to be contnuous. The crcle group can be thought of as another n lmt of the C n, hence the Z n, group. Ths s a dfferent lmt than the group of ntegers Z. 2.6 Matrx group A set of matrces can form a group. General Lnear Matrx Group: the set of n n nvertble matrces wth matrx multplcaton as the composton rule forms a group. Comments: 2
3 (1) If the entres of the matrces are real numbers, the matrx group s sad to be over R and denoted as GL(n, R). If the entres of the matrces are complex numbers, the matrx group s sad to be over C and denoted as GL(n, C). (2) The dentty element n the group s the dentty matrx. (3) The matrx group s n general nonabelan. (4) If we restrct to the set of matrces wth determnant one, we get the specal lnear group SL(n, R) or SL(n, C). (5) We can also restrct to orthogonal or untary matrces and get the orthogonal group O(n) or the untary group U(n). 3 Basc concepts n group theory 3.1 Conjugacy class Conjugacy, defnton: two elements a and b of a group G are conjugate f there exsts an element g G such that a = gbg 1. The element g s called the conjugatng element. Example: n the Dhedral group D 3, two reflectons b 1 and b 2 are conjugate because b 2 = cb 1 c 1. Propertes: (1) every element s conjugate to tself a = eae 1. (2) f a s conjugate to b (a = gbg 1 ), then b s conjugate to a (b = g 1 ag). (3) f a s conjugate to b (a = gbg 1 ), and b s conjugate to c (b = hch 1 ), then a s conjugate to c (a = ghc(gh) 1 ). Conjugacy defnes a partcular knd of equvalence relaton among group elements and conjugate elements are smlar to each other n some ways. For example, f a and b are conjugate to each other, then they have the same order. To show ths, assume the order of a s k a and the order of b s k b and b = gag 1. Then b ka = (gag 1 ) ka = ga ka g 1 = geg 1 = e (2) Therefore, k b s a dvsor of k a. Smlarly we can show that k a s a dvsor of k b. Therefore, k a = k b. Conjugacy class: Elements of a group whch are conjugate to each other are sad to form a conjugacy class. Comments: 3
4 (1) Each element of a group belongs to one and only one conjugacy class. That s, dfferent conjugacy classes are dsjont. (If a s conjugate wth a set of b s and also conjugate wth a set of c j s, then the b s and c j s are also conjugate wth each other and they belong to the same conjugacy class.) (2) The dentty element forms a class by tself. (For any g G, geg 1 = e.) (3) Each group can be parttoned nto a number of dsjont conjugacy classes. Examples: (1) Cyclc group C n Because gag 1 = a for any a and g n C n, each group element forms a conjugacy class by tself. The number of conjugacy classes s equal to the number of group elements. Ths s a result common to all abelan groups. (2) Dhedral group D 3 C A b 3 b 1 c b 2 B Fgure 2: D 3 as the symmetry group of undrected regular trangle D 3 contans 6 group elements {e, c, c 2, b 1, b 2, b 3 }, where c s rotaton by 2π/3 and b s are reflecton operatons. Drect calculaton shows b cb 1 = c 2, cb c 1 = b (mod 3)+1, (3) That s, c s conjugate to c 2, and the b s are conjugate to each other. Therefore these 6 elements can be parttoned nto three conjugacy classes (e), (c, c 2 ), (b 1, b 2, b 3 ). Elements n the same conjugacy class represent smlar operatons: dong nothng, rotaton, reflecton. (3) Dhedral group D 4 D A C b 4 c B b 1 b 2 b 3 Fgure 3: D 4 as the symmetry group of undrected square 4
5 The D 4 group contans 8 elements {e, c, c 2, c 3, b 1, b 2, b 3, b 4 } (4) where c s rotaton by π/2, and b s reflecton across the correspondng axs. Drect calculaton shows that cb 1 c 1 = b 3, cb 3 c 1 = b 1, cb 2 c 1 = b 4, cb 4 c 1 = b 2, b cb 1 = c 3 (5) Therefore, the 8 elements are parttoned nto fve conjugacy classes (e), (b 1, b 3 ), (b 2, b 4 ), (c, c 3 ), (c 2 ). Note that whle elements n the same conjugacy class have the same order, the reverse s not true. For example, b 2 and b 1 are both order 2 elements, but they are not conjugate to each other. Conjugaton has a very physcal meanng. gag 1 s mplementng the symmetry transformaton of g on the symmetry transformaton of a. For example, cb 1 c 1 corresponds to rotatng the reflecton operaton of b 1 by π/2. As we can see, f we rotate the reflecton axs of b 1 by π/2, we get b 3. Indeed, our prevous calculaton shows that cb 1 c 1 = b 3. Another example s b cb 1 whch mplements reflecton on the elementary rotaton operaton and map t to the nverse rotaton operaton. Indeed our calculaton shows that b cb 1 = c 3. 5
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