解 [tex=26.929x2.786]1nvd38z/cgkkKYpklmo2Tc7pHn38rDVupSknsc5iGgW6QsHWh69Mv7mca4tn0wCRuhDXoE1g23pgO/AueY4FZUthdgnLpA/V53OCELPz8B8qlYgGV5Uvd9JrBXvBeKPOyb7quPiap8/furOOblP12v0FXOou/IeK6Pan/KYo60oOuFtlGHRA5kMLQo0JlZBoDxQYJED9cBtv8FAXyOXeD7uicTzVZsOFbhpOwHX9gZEu+LdtptcRAQuJ+5zK/rkcK72E2DCJ+rLkRWd3fef+rnxDMHwCu5su185SWDCezWe3fd03HOEbmn1cSWQAzXxBoNjTLjvuV/0qxtfOu65NYw==[/tex][tex=25.5x2.786]TOK5a2zbqX3s0g/2gMYj+bvH1L5vOxEKWtX1C1oNxsscIUXQIgkhay8gSODcqbXuJD/xELX3/f//NzFP5ylKOOCHi4i+fsMIxJBDBP9HD9EmeDuWg8FCQY1FHggZYE91nU2KnEKftTM1ACI2VC9UjRVmeeO/gbjbsNG2r6PEPIITm0sqn839YtLYIVeoV0mPD9L44myjMmVJ8+1KvBAZdmo7cwZe0i3qOh2bujyA7yo1TOFId+X02A2y4GCASW/2C+fhPw8iPCTXvvtJHzE+xZmq8crD0BeTpMRAHp46R7CBk9uvP0n5GJGD8enIeaUfFRQEpDP7w8AD7Ktfsdxnnw==[/tex]。为便于叙述，下面引进群中元素的阶的概念。群中元素[tex=0.571x0.786]HXNXn3AXpwdIpZt8+6oCEw==[/tex]的阶是指适合 [tex=2.286x1.0]qF9xj8o2mMMaUsMgaHkC6g==[/tex]( 单位元 )的最小正整数[tex=0.643x0.786]h6IfGOxBlahC8le5jX4WiA==[/tex]。从定义不难看出，[tex=2.5x1.143]WVLtR3VPdSlf6qWJAi7er9WdTCDoM59RqvyklV6jSsU=[/tex]时，[tex=0.5x0.786]WKYr2kz69xrVCyPvbyVG1w==[/tex]，[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]，[tex=1.286x0.786]lRSLJav0cvc1uYdx/9plcw==[/tex]，[tex=1.929x1.214]8zmgPbemfkRq48xw/z9enQ==[/tex]互不相等。否则，存在整数[tex=0.571x1.0]CQkpoDeAAI+5FKIfe1wVCA==[/tex]，[tex=0.357x1.0]5vVfAZliYwqMw8JaLE+iEA==[/tex]，[tex=7.214x1.143]GqVU0tOG29aG5I/eO/efBm7xl4tT4+ObGm1GW3WXyjc=[/tex]，使[tex=2.5x1.214]gPq4oH8nANlZpkC+8Pj00g==[/tex]，于是 [tex=13.071x1.786]cFVljmEvKfHdaQPGG2Czras/jkbA6/DQE8fsdPvZilkODsx90YUyRh8Hqy9JQd1MQT0tZMd5cqS+de0ln0+0OjzTCwS9ld8zJvXdxInAv0IwQTDtAfjSzpzMz+TnTfB4[/tex]。显然 [tex=5.429x1.143]1TqBjmJpRPUWnuN/PBCUJw==[/tex]，与[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]的阶是[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]矛盾。由此知循环群[tex=1.357x1.357]UMu6yZaqu6lAbCVsfR7R0Gd4uDjR1gRFcqTenXrRKBI=[/tex]为[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶子群。[tex=1.0x1.214]vIC1ui1s5j6wm/e+z3rn5A==[/tex]中[tex=0.5x0.786]xdTs2QHMXTpKzI7ZnwCRMQ==[/tex]为一阶元，[tex=0.929x1.0]GIJA+ElcF8sfFVDzHxfsvg==[/tex]，[tex=0.929x1.0]gak5irftAp3YubasKVoCqw==[/tex]，[tex=0.929x1.0]G+/HNITXSFKSvAAtLR9AqA==[/tex]为二阶元，[tex=0.929x1.0]vjXVNjFU7EbPD0ok8JKRfA==[/tex]，[tex=0.929x1.0]c1LK6GWhoinqYyFdR5Ltvw==[/tex]为三阶元，因而有子群[tex=5.429x1.357]1DveO5sagClwV7/15tlBrUj9NrGPxcj2v9pCwRPbGxVhp2JuC9X1EXQhlFK94Uz7[/tex]，[tex=7.286x1.357]Zl8AgQ+y5fHkXSFXHXV35VN9jmm2Yes2VGTjcXxZ6ru6ZBTqPVlBSU/4dIUfaAH0[/tex]，[tex=7.643x1.357]Du0xpm+XEihVAwB/EM2ATGKQrfRw5VENHEPXBbc3mK8rkC2tMz/hkkLKzpg9g32l/68WQ9gh7fohn32H2Ua3184icYAcl0WyQzNlQy1Ee5U=[/tex][tex=7.286x1.357]m/euSIzel1jd/R7dhUam08/YoTMnZ84Jr36QZinFEKIQYHwfafpBm9P1x4pYby/RCKqD7fK4oy1Z6INkfMzbYg==[/tex]，[tex=11.143x1.357]CeFuG+lZIXAEDzesAF95i89ihPOH3jlOVkBHdweomMyPqygODlxJgB1FnH7rEbFrZ7ZDx2Cp2S1RGQFkX7Cn7XJLUyNmFVOQAeZYZXPNQV79x2c5lqVZJbxCJ9oYXswq[/tex]，其中[tex=1.214x1.214]QhT2/HdtDYsWF2BKKz3ziQ==[/tex]的构成是由[tex=2.786x1.5]QJi1+k3CwswPDSl7A53vPAAQoO8Xw3xJqnWQsi3b0f8=[/tex]，[tex=2.714x1.5]F2lDOoHRKeTM+XJtV95FsjgRzJ0ooHeOTGEcJigKE84=[/tex]， [tex=4.0x1.5]oJKBhDIs8Ig4u104mX1FSSIkB0JPGEX+w7H8zZTkrdo=[/tex]所致。若有子群[tex=0.857x1.0]aPLFPHMGSKDwulHSwLWugg==[/tex]包含一个二阶元[tex=0.857x1.0]8x4jXMFIAZ2p93jtl+14DA==[/tex]及一个三阶元[tex=0.929x1.071]Emuwhfx0zByZ+K3DX88iOQ==[/tex]由直接验证可知[tex=4.786x1.286]QNoLiZOYM6kkcILBWMvDKvm1v/rF+BBiADmRYlApqp1L37FklbgimV2jYQI4Ryvm[/tex]，并且[tex=1.714x1.071]4CmxFD5OyIQ7gEBnNit/pAMehQelsaIKXwcU0vrqrlE=[/tex]，[tex=1.714x1.071]ATWpsw2jZqS3B8ffGVHO+p0UGYYD1y78T5mTC0DV+WY=[/tex]都是与[tex=0.857x1.0]8x4jXMFIAZ2p93jtl+14DA==[/tex]不同的二阶元，故[tex=1.0x1.214]vIC1ui1s5j6wm/e+z3rn5A==[/tex]中所有三个二阶元[tex=0.929x1.0]GIJA+ElcF8sfFVDzHxfsvg==[/tex]，[tex=0.929x1.0]vjXVNjFU7EbPD0ok8JKRfA==[/tex]，[tex=2.429x1.214]xDcXAAsdhBRRH9ePgnHRmw==[/tex]。又[tex=2.5x1.071]kb+F1tkyGadGcF6nmVL+vlvoNUjC39hVok7QGThutOm97SqWvhXbOo2Czo7257Sb[/tex]是不同于[tex=0.929x1.071]7hT1oGaOt3ItjD7Cvc0/WA==[/tex]的三阶元，所以[tex=1.0x1.214]vIC1ui1s5j6wm/e+z3rn5A==[/tex]中所有的两个三阶元 [tex=0.929x1.0]GIJA+ElcF8sfFVDzHxfsvg==[/tex]， [tex=2.429x1.214]eEJrCtKUu1TlZWmKuzfhRA==[/tex]从而[tex=2.571x1.214]ZVKFQvXi6ivl/k/I8zAiEw==[/tex]。若有子群含有两个二阶元，则同样可证[tex=2.643x1.214]CWFzp+T+SnkN5yG7ebDg2g==[/tex]。故 [tex=1.0x1.214]vIC1ui1s5j6wm/e+z3rn5A==[/tex]的一切子群恰是[tex=1.214x1.214]+qJ4vkt58gzxUz3UM+z5ww==[/tex], [tex=1.214x1.214]Lkp5hpVm9tm9cgUSIXEPLg==[/tex]，[tex=1.214x1.214]ba06lGwptRjJwZ2CKYOg1g==[/tex]，[tex=1.214x1.214]51w4NzjuTZYAVm7PZZrO4Q==[/tex]，[tex=1.214x1.214]QrsUoOgj+1+V02P/noUGfg==[/tex]，[tex=1.0x1.214]9PE628d4jwg8J3nqnd5uOg==[/tex]。