Re: [代數] 代數(2)

[Zebulon (妳對我的好)]

:   If G is a group of order 2k, where k is odd,
:      then G has a subgroup H of order k.
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|G| = 2k  , k is odd.
A(G) :={f:G ─> G | f is a bijection map (1-1 and onto)}
The operation on A(G) is the composition of functions.
Then A(G) is a group which is isomorphic to S(2k)

For each g in G, let  Tg: G ─> G  Tg(x):= gx

1. ι:G ─> A(G)  ι(g) :=Tg
     is a well-defined group monomorphism (1-1 homomorphism)
         check: a. Tg is  in A(G)
                b. ι is a group homomorphism
                c.  Ker(ι) = {e} (the identity in G)
     Hence G is embeded into A(G) as a subgroup ι(G) of A(G).

2. identify  A(G) and S(2k).
        If α in G is a element of order 2,
        then ι(α) is a odd permutation.
         (check the number of transpositions of it )

        Therefore the set of all even permutations of ι(G)
        is a subgroup of ι(G) of order |ι(G)|/2 = k.
        The preimage of it is a subgroup of G of order k.

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證明好像很長  不知道有沒有短一點的

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◆ From: 210.85.44.161
推 Sfly:G --> S(2k) 的embeding是well-know的(Cayley thm)
→ Sfly:所以考慮α的orbit, α是一些不相交對換的積
→ Sfly:but α doesn't fix any element of G,so α is odd