Đây là chứng minh cho định lý đã thảo luận với
nemo. Chứng minh này có trong cuốn của Rotman
Theorem 1: If the order of
http://dientuvietnam...n/mimetex.cgi?G is
http://dientuvietnam...imetex.cgi?p^sm, then
http://dientuvietnam...ex.cgi?p|(N_r-1), where
http://dientuvietnam...mimetex.cgi?N_r is the number of subgroups of order
http://dientuvietnam...mimetex.cgi?p^r in
http://dientuvietnam...n/mimetex.cgi?G, and
http://dientuvietnam...ex.cgi?0<r<s 1.Proof: We can assume
http://dientuvietnam...mimetex.cgi?s>1 and prove by induction.
For
http://dientuvietnam...imetex.cgi?r=1: (Hint) Let
http://dientuvietnam...mimetex.cgi?Z_p act on http://dientuvietnam.net/cgi-bin/mimetex.cgi?X={(g_1,g_2,...,g_p)|g_1g_2...g_p=1}
For http://dientuvietnam.net/cgi-bin/mimetex.cgi?r>1:
Fix a subgroup http://dientuvietnam.net/cgi-bin/mimetex.cgi?H of order http://dientuvietnam.net/cgi-bin/mimetex.cgi?p^{r-1}, let http://dientuvietnam.net/cgi-bin/mimetex.cgi?K_1,...,K_a be subgroups of order http://dientuvietnam.net/cgi-bin/mimetex.cgi?p^r in http://dientuvietnam.net/cgi-bin/mimetex.cgi?G such that http://dientuvietnam.net/cgi-bin/mimetex.cgi?H is contained in http://dientuvietnam.net/cgi-bin/mimetex.cgi?K_i, so we know that H is a normal subgroup of http://dientuvietnam.net/cgi-bin/mimetex.cgi?K_i.
Thus, http://dientuvietnam.net/cgi-bin/mimetex.cgi?K_i is contained in http://dientuvietnam.net/cgi-bin/mimetex.cgi?N_G(H), the normalizer of http://dientuvietnam.net/cgi-bin/mimetex.cgi?H in http://dientuvietnam.net/cgi-bin/mimetex.cgi?G.
So http://dientuvietnam.net/cgi-bin/mimetex.cgi?a equal the number of subgroups of order http://dientuvietnam.net/cgi-bin/mimetex.cgi?p in http://dientuvietnam.net/cgi-bin/mimetex.cgi?N_G(H)/H, and http://dientuvietnam.net/cgi-bin/mimetex.cgi?a is congruent to http://dientuvietnam.net/cgi-bin/mimetex.cgi?1 mod http://dientuvietnam.net/cgi-bin/mimetex.cgi?p.
Fix a subgroup http://dientuvietnam.net/cgi-bin/mimetex.cgi?K of order http://dientuvietnam.net/cgi-bin/mimetex.cgi?p^r, let http://dientuvietnam.net/cgi-bin/mimetex.cgi?H_1,...,H_b be subgroups of order http://dientuvietnam.net/cgi-bin/mimetex.cgi?p^{r-1} in http://dientuvietnam.net/cgi-bin/mimetex.cgi?G such that http://dientuvietnam.net/cgi-bin/mimetex.cgi?K contains in http://dientuvietnam.net/cgi-bin/mimetex.cgi?H_j, so we know that http://dientuvietnam.net/cgi-bin/mimetex.cgi?H_j is a normal subgroup of http://dientuvietnam.net/cgi-bin/mimetex.cgi?K.
Thus, http://dientuvietnam.net/cgi-bin/mimetex.cgi?H_1H_2=K and http://dientuvietnam.net/cgi-bin/mimetex.cgi?|D|=p^{r-2}, where http://dientuvietnam.net/cgi-bin/mimetex.cgi?D is the intersection of http://dientuvietnam.net/cgi-bin/mimetex.cgi?H_1 and http://dientuvietnam.net/cgi-bin/mimetex.cgi?H_2. Therefore, http://dientuvietnam.net/cgi-bin/mimetex.cgi?|K/D|=p^2 and http://dientuvietnam.net/cgi-bin/mimetex.cgi?K/D is abelian.
http://dientuvietnam.net/cgi-bin/mimetex.cgi?K/D ~ http://dientuvietnam.net/cgi-bin/mimetex.cgi?K/D has http://dientuvietnam.net/cgi-bin/mimetex.cgi?p+1 subgroups of order http://dientuvietnam.net/cgi-bin/mimetex.cgi?p, and these subgroups correspond one-to-one with the subgroups of order http://dientuvietnam.net/cgi-bin/mimetex.cgi?p^{r-1} contained in http://dientuvietnam.net/cgi-bin/mimetex.cgi?A and containing http://dientuvietnam.net/cgi-bin/mimetex.cgi?D.
If http://dientuvietnam.net/cgi-bin/mimetex.cgi?H_j is not in this list
http://dientuvietnam.net/cgi-bin/mimetex.cgi?D is not contained in http://dientuvietnam.net/cgi-bin/mimetex.cgi?H_j, let http://dientuvietnam.net/cgi-bin/mimetex.cgi?E be the intersection of http://dientuvietnam.net/cgi-bin/mimetex.cgi?H_1 and http://dientuvietnam.net/cgi-bin/mimetex.cgi?H_j. Then we get another list of http://dientuvietnam.net/cgi-bin/mimetex.cgi?p+1 subgroups of order http://dientuvietnam.net/cgi-bin/mimetex.cgi?p^{r-1} and contain http://dientuvietnam.net/cgi-bin/mimetex.cgi?E in http://dientuvietnam.net/cgi-bin/mimetex.cgi?K, and this list intersects the old list only at http://dientuvietnam.net/cgi-bin/mimetex.cgi?K_1(=E.D).
So http://dientuvietnam.net/cgi-bin/mimetex.cgi?b equal the number of subgroups of order http://dientuvietnam.net/cgi-bin/mimetex.cgi?p^{r-1} in http://dientuvietnam.net/cgi-bin/mimetex.cgi?K, and http://dientuvietnam.net/cgi-bin/mimetex.cgi?b is congruent to http://dientuvietnam.net/cgi-bin/mimetex.cgi?1 mod http://dientuvietnam.net/cgi-bin/mimetex.cgi?p.
We also know that http://dientuvietnam.net/cgi-bin/mimetex.cgi?\Bigsum_{i=1}^{n_{r-1}}{a_i}=\Bigsum_{j=1}^{\n_r}{b_j}, so http://dientuvietnam.net/cgi-bin/mimetex.cgi?n_r is congruent to http://dientuvietnam.net/cgi-bin/mimetex.cgi?n_{r-1} mod http://dientuvietnam.net/cgi-bin/mimetex.cgi?p, and by the induction hypothesis, http://dientuvietnam.net/cgi-bin/mimetex.cgi?n_r must be congruent to http://dientuvietnam.net/cgi-bin/mimetex.cgi?1 mod http://dientuvietnam.net/cgi-bin/mimetex.cgi?p.
Theorem 2: If the order of http://dientuvietnam.net/cgi-bin/mimetex.cgi?G is http://dientuvietnam.net/cgi-bin/mimetex.cgi?p^s, then http://dientuvietnam.net/cgi-bin/mimetex.cgi?NN_r is the number of normal subgroups of order http://dientuvietnam.net/cgi-bin/mimetex.cgi?p^r in http://dientuvietnam.net/cgi-bin/mimetex.cgi?G, and http://dientuvietnam.net/cgi-bin/mimetex.cgi?0<r<s+1.
Proceed similarly to the proof above, except that in the case http://dientuvietnam.net/cgi-bin/mimetex.cgi?|K|=p^r, we let http://dientuvietnam.net/cgi-bin/mimetex.cgi?G act on the set http://dientuvietnam.net/cgi-bin/mimetex.cgi?X={H_1,...,H_b} by conjugation action.
http://dientuvietnam.net/cgi-bin/mimetex.cgi?b=|X| is congruent to the http://dientuvietnam.net/cgi-bin/mimetex.cgi?|X^G| (mod http://dientuvietnam.net/cgi-bin/mimetex.cgi?p), where http://dientuvietnam.net/cgi-bin/mimetex.cgi?X^G is the number of points in http://dientuvietnam.net/cgi-bin/mimetex.cgi?X that are fixed by http://dientuvietnam.net/cgi-bin/mimetex.cgi?g for all http://dientuvietnam.net/cgi-bin/mimetex.cgi?g in http://dientuvietnam.net/cgi-bin/mimetex.cgi?G.
We also know that http://dientuvietnam.net/cgi-bin/mimetex.cgi?X^G is the set of subgroups http://dientuvietnam.net/cgi-bin/mimetex.cgi?H of order http://dientuvietnam.net/cgi-bin/mimetex.cgi?p^{r-1} in http://dientuvietnam.net/cgi-bin/mimetex.cgi?K such that http://dientuvietnam.net/cgi-bin/mimetex.cgi?H is normal in http://dientuvietnam.net/cgi-bin/mimetex.cgi?G. The theorem thus follows.
Bài viết đã được chỉnh sửa nội dung bởi nemo: 30-10-2005 - 09:12