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作者  kyod ( ) 站內  ALGEBRA
標題  [問題] cyclic group
時間  2009/11/02 Mon 00:35:06


Let a,b be elements of a group G.

show that │a│=│a^-1│

          │ab│=│ba│

          │a│= │cac^-1│              where │a│=│a^-1│


(pf.)

(1)

Suppose that <a>:The cyclic group which generated by element a
             <a^-1>:The cyclic group which generated by element a^-1

Claim:<a> = <a^-1>

∵ The element of <a> is a^n for some nεZ

a^n = (a^-1)^(-n) => the element of <a^-1> is (a^-1)^(-n) for some nεZ

=> <a> = <a^-1>

=> │a│=│a^-1│

(2)

Case1:The order of ab is finite i.e │ab│= n < ∞

∵ <ab> is a cyclic group & │ab│= n
       n
=> (ab)  = e

=> abab....abab = e

=> (a^-1)*(abab....abab)*a = (a^-1 * a)*(baba.....baba) = e
       n
=> (ba)  = e

∴│ab│=│ba│

Case2:The order of ab is infinite

∵ <ab> is a cyclic group & │ab│= ∞

∴ <ab> is isomorphic to < Z , + >

Similarly, <ba> is isomorphic to < Z , + >

∴ <ab> is isomorphic to <ba>

∴│ab│=│ba│
                    n     n
(3) Claim: (cac^-1)  = ca c^-1

By induction on n ,
                      0         0
n=0 , L.H.S = (cac^-1)  = e = ca c^-1 = R.H.S (ok)
                                                         k     k
Suppose that the statement is ture for n=k , i.e (cac^-1)  = ca c^-1

Now , for n = k+1 ,
        k+1          k              k                   k
(cac^-1)   = (cac^-1) (cac^-1) = (ca c^-1)*(cac^-1) = ca (cc^-1)c^-1
                k+1
           =  ca   c^-1

∴ The statement is true for nεN∪{0}

Similarly , the statement is true for n which is negative integer.
          n     n
∴(cac^-1)  = ca c^-1 for all nεZ

Suppose that │a│= n < ∞
      n
i.e  a = e
                       n     n
Now , consider (cac^-1)  = ca c^-1 = cec^-1 = e for all nεZ

i.e │cac^-1│= │a│= n
                         #

我的問題在於,如果order是finite應該都沒有太大的問題可以證得出來

可是當order是infinite的時候,就不知如何下手

可以請各位大師指點一下嗎?

感謝。

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★ kyod  好吃好吃!!!(猛點頭中)
★ kyod:妳知道為什麼好吃嗎???
★ kyod  不知道耶!!!(笑笑地搖搖頭聳聳肩)
★ kyod:因為有我在呀!!....哈哈...
★ kyod  討厭啦....(害羞中)

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  •FROM [kyod 從 140.122.175.82 發表]
□ Modify: 2009/11/02 Mon 13:15:47  140.122.140.211 修改
→ gawk :第三個  let x=c^{-1}  y=ca then by 第二個                 09/11/02
→ gawk :第二的case 2  我覺得應該要證if |ab|=infty  then |ba| must 09/11/02
→ gawk :be infty                                                  09/11/02
→ gawk :你的證明看起來好像|ab|=infty  和|ba|=infty是已知          09/11/02

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