作者Zebulon (妳對我的好)
看板Math
標題Re: [代數] 代數(2)
時間Mon Jul 25 00:31:28 2005
: 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.
~~
證明好像很長 不知道有沒有短一點的
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1F：推 Sfly:G --> S(2k) 的embeding是well-know的(Cayley thm)220.135.132.108 07/25
2F：→ Sfly:所以考慮α的orbit, α是一些不相交對換的積220.135.132.108 07/25
3F：→ Sfly:but α doesn't fix any element of G,so α is odd220.135.132.108 07/25