[代數] Cosets

[air11 (拿出一張白紙...)]

1. In the additive group |R^m of vectors, let W be the set of solutions
   of a system of homogeneous linear equations AX=0. Show that the set
   of solutions of an inhomogeneous system AX=B is either empty or
   it is an (additive) coset of W

   首先這題的A應該是個方陣吧?!
   那麼AX=B無解的話，表示set of solution of AX=B is empty.
   若是以這樣的想法，我想不出如果有solution的話，和coset會有甚麼關係.....


2. A group G of order 22 contains elements x and y, where x≠1 and y is not
   a power of x. Prove that the subgroup generated by these elements in the
   whole group G.

   依照題意我覺得會用到cyclic group，但是不知道從何下手......



  感謝各位大大指教 :)

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◆ From: 125.225.15.246
推 jacky7987   :AX=B 有一個解叫做v的話那 v+W都是解
→ jacky7987   :因為A(v+W)=Av+0=B
推 zombiea     :2: |x|=2, 11, 22, for first 2 cases, we have
→ zombiea     :G=<x,z> for some z, then y=x^az^b, b=\=0, rewrite
→ zombiea     :z by x and y...
→ zombiea     :here note that G also equal to <x,z^b> for 2, 11
→ zombiea     :are primes.
→ air11       :感謝以上兩位大大的提示，我在想想看 :D