[代數] 兩題Sylow subgroup的問題~求提示

看板Math作者 (映雪)時間13年前 (2013/02/09 11:54), 編輯推噓2(2推 0噓 14→)
留言16則, 4人參與, 6年前最新討論串1/1
如標題,題目如下: 1. If P is a normal Sylow p-subgroup of a finite group G, show that α(P)包含於P for every homomorphism α:G→G. 2. Let P is a normal subgroup of H, H is a normal subgroup of G. If P is a Sylow subgroup of G, show that P is a normal subgroup of G. 謝謝大家~~ m_ _m 祝各位蛇年行大運 =D -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 114.40.193.58
推
02/09 12:26, , 1F
1. The restriction of α on P is still
02/09 12:26, 1F
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02/09 12:28, , 2F
a homomorphism. By Lagrange's thm., α(P) is of
02/09 12:28, 2F
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02/09 12:30, , 3F
order some power of p. Then α(P) lies in some
02/09 12:30, 3F
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02/09 12:30, , 4F
Sylow p-subgroup by Sylow 1st.
02/09 12:30, 4F
推
02/09 12:44, , 5F
2.設 P 是 G 的 Sylow p-subgroup, 則也是 H 的Sylow
02/09 12:44, 5F
→
02/09 12:45, , 6F
p-subgroup. 因為 P normal in H, 所以 P 是 H 唯一
02/09 12:45, 6F
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02/09 12:45, , 7F
的 Sylow p-subgroup. 因為 H normal in G, 所以對
02/09 12:45, 7F
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02/09 12:46, , 8F
任意 x in G 有 xHx^{-1} \subseteq H. 得 xGx^{-1}
02/09 12:46, 8F
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02/09 12:47, , 9F
\subseteq H => xGx^{-1} 是 H 的 Sylow p-subgroup
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02/09 15:08, , 10F
最後幾行有點亂？好像應該是xPx^{-1} XD
02/09 15:08, 10F
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02/09 15:38, , 11F
對耶XDD
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08/13 17:26, , 12F
1. The rest https://noxiv.com
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09/17 15:20, , 13F
a homomorph https://daxiv.com
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11/10 11:24, , 14F
//noxiv.com https://muxiv.com
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01/02 15:16, 7年前 , 15F
01/02 15:16, 15F
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07/07 10:38, 6年前 , 16F
//muxiv.com http://yaxiv.com
07/07 10:38, 16F
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