[代數] 幾題關於group的證明@@

[paggei (XD)]

以下(G,*,e)為一群，想證                         j
                     -1   r        j      -j   r
(1) if a, b∈G, b*a*b  = a , then b * a * b = a   for some r∈Ｎ

    這題想不出來啊...想用數學歸納法，                          k
         k+1    -(k+1)          k        -k  -1               r
    可是b   *a*b       頂多拆b(b  * a * b  )b  然後中間卡一個a  ....

(2) │G│ is finite, H is a subset of G, prove that H is a subgroup of G
     if and only if e∈H and a*b∈H for all a, b∈G

     不會證inverse∈H的部分@@
(3) H, K are subgroup of G, prove that if H∪K is a subgroup of G,
    then H⊆K or K⊆H

--

--
※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 114.41.203.186
→ alasa15     :第二題是for all a, b∈H吧?
→ alasa15     :∵│G│is finite ∴{a^i}must be cyclic
→ alasa15     :i.e. there is n∈N s.t. a^n = e
→ alasa15     :pick a^(-1) = a^(n-1)
→ alasa15     :note that a∈G
→ paggei      :對 是屬於H沒錯@@ 打錯
→ paggei      :感謝@@