几个有趣的同构结论Theorem 9.3 IfG is a cyclic group of order n, then G is isomorphic to ZnCorollary 9.4 If G is a group of order p, where p is aprime number,thenGis isomorphicto Zp.Theorem 9.6 (Cayley) Every group is isomorphic to agroup of permutations
几个有趣的同构结论
Carley定理的证明Theorem 9.6 (Cayley) Every group is isomorphic to a group of permutations·从任意一个群G出发，构造一个置换群G，：G={入g:gEG])·由置换函数组成的群·由G出发，构造置换函数，置换函数的个数和群G相同入g(a) = ga.·构造群G到置换群G’的同构函数Φ: g入g·证明这个函数的双射·证明这个函数是G到G’的同构
Carley定理的证明 • 从任意一个群G出发，构造一个置换群G’： • 由置换函数组成的群 • 由G出发，构造置换函数，置换函数的个数和群G相同 • 构造群G到置换群G’的同构函数 • 证明这个函数的双射 • 证明这个函数是G到G’的同构
问题4：为什么下面的结论不叫“定理”？Proposition 9.7 Let G and H be groups. The set G x H is a group underthe operation (g1, hi)(g2, h2) = (g192,hih2) where g1,g2 E G and h1, h2 E H.Example 9. The group Z, considered as a set, is just the set of all binaryn-tuples.The group operation is the“exclusive or"of two binary n-tuples.For example,(01011101)+(01001011)=(0Q010110)问题5.2：这个操作从何而来？问题5.1：这个符号是什么意思？
问题4：为什么下面的结论不叫“定理”？ 问题5.1：这个符号是什么意思？ 问题5.2：这个操作从何而来？
不难理解的几个定理：Theorem 9.8 Let (g.h)EGx H.If gandh have finite orders r and srespectively,then the order of (g,h) inG× H is the least common multipleofrands.Corollary 9.9 Let (gi,..., gn) eIGi.If gi has finite order ri in Gi, thenthe order of (g1,..., gn) in IIG, is the least common multiple of ri, ..., Tn.问题6：如果诸ri互素，会有什么结论？
不难理解的几个定理： 问题6：如果诸ri互素，会有什么结论？