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1、 I K 8 JJII JI 11 ?48 w 4 1l + kerf = mZ, m 0 , Z/kerf )?k, (k,m) = 1. d, n. y I K 8 JJII JI 19 ?48 w 4 nnn1 G k?+. ?a,b G, e(ord(a),ord(b) = 1, K ord(ab) = ord(a)ord(b). yyy aord(ab)ord(b)= (ab)ord(ab)ord(b)= 1, ord(a) | ord(ab)ord(b). ? ord(a) | ord(ab). n, ord(b) | ord(ab). ord(a)ord(b) | ord(ab)
2、. d?, w,k ord(ab) | ord(a)ord(b). ord(a)ord(b) | ord(ab). d, ord(ab) = ord(a)ord(b). I K 8 JJII JI 110 ?48 w 4 nnn2 G k?+. ?a,b G, 3c G ord(c) = ord(a),ord(b). yyy d1.5 5, uord(a) ord(b), 3u, v v: u|ord(a),v|ord(b),(u,v) = 1 ord(a), ord(b) = uv. y3-s = ord(a) u ,t = ord(b) v , n3(viii), k ord(as) =
3、ord(a) (ord(a),s) = u,ord(bt) = v. 2n1, ord(asbt) = ord(as)ord(bt) = uv = ord(a), ord(b). d, c = asbt. =. y I K 8 JJII JI 111 ?48 w 4 nnn6G k ? +,KG 3 ? a1,a2,.,as g ? m1, m2, .,msvmi|mi+1, 1 i s 1, G = . y G = b1, b2, ., bn. dn2, 3?c1 ord(c1) = ord(b1),.,ord(bn). -H1=, K G/H1= b11H1,.,b1n1H1,n1= G
4、: H1 l, kl(i(1) 1 ) = i(1) 1 . k1 n k1(i(1) 1 ) = i(1) 1 ?, - i(1) 2 = 1(i(1) 1 ), ., i(1) k1 = k11(i(1) 1 ). K1= (i(1) 1 , i(1) 2 , ., i(1) k1 ) k1. XJk1= n, K = 1. (. I K 8 JJII JI 129 ?48 w 4 XJk1, | | | | |, 1 i 2. I K 8 JJII JI 142 ?48 w 4 13. n, (Z/nZ)4 + +, | | | | |, 1 i 3 14. y: +S4|)? (1,2
5、),(1,3),(1,4). ?, T|)?5S4kf+. 15. y:GL2(Z) = a b c d | a,b,c,d Z, ad bc = 1 u ?+. T = 1 1 0 1 ,S = 01 1 0 GL2(Z) |)?. I K 8 JJII JI 143 ?48 w 4 I K 8 JJII JI 144 ?48 w 4 I K 8 JJII JI 145 ?48 w 4 1. y: XJa,b +G ?, K (ab)1= b1a1. 2. y: +G ?+?a,b G , k(ab)2= a2b2. 3. y: +G ?+?a,b G , k (ab)3= a3b3,(ab
6、)4= a4b4,(ab)5= a5b5. 4. G n k+. y: ?a G, kan= e. 5. y: +G ?a _?a1k. I K 8 JJII JI 146 ?48 w 4 6. G +. P cent(G) = a G | ab = ba b G. y: cent(G) G 5f+. 7. a +G ?. y: N : x 7 axa1 G gg?. 8. H +G f+. 3G XR: aRb X Jb1a H. y: (i) R dX. (ii) aRb aH = bH. 9. z+?+. 10. F7LL. I K 8 JJII JI 147 ?48 w 4 11. y
7、F23“?u?+, )?. 12. y: Z/nZ _?+, P Z/nZ. 13. y: Z/26Z ?+. 14. ?E26 ?+. I K 8 JJII JI 148 ?48 w 4 10. y: SL2(Z) = ( a b c d ! | a,b,c,d Z,ad bc = 1 ) +, )? T = 1 1 0 1 ! , S = 01 1 0 ! yyy 1 1 0 1 !q = 1 q 0 1 ! 01 1 0 ! 01 1 0 ! = 10 01 ! a b c d ! 01 1 0 ! = b a d c ! ?|c| |d| , |d?c,d . a b c d ! 1 q 0 1 ! = a aq + b c cq + d ! ?|c| |d| , |d$d .
