Re: [代數] 用order來判斷group的性質

看板Math作者yusd24 (阿鄉)時間15年前 (2011/01/23 00:53)推噓0(0推 0噓 9→)

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※ 引述《wyob (Go Dolphins)》之銘言： : 最近在複習代數時做題目的時候，發現這類型的題目 : 覺得能用order來判斷群很有趣於是就多找了一些題目 : 在還沒讀到有關Sylow定理前有些這類型的題目，能用其他的性質就判斷出來 : 可是像order大一點的就不會了，所以想來這請教幾題 : 1.If o(G)=15 prove G is cyclic group(這題如果不用sylow定理怎麼做?) : 2.o(G)=pq where p,q be prime and p<q,prove if p不能整除(q-1) then G : is cyclic group(這如果是對的就能證明上面那題，不過不知怎麼證這題) Let H be the q-Sylow of G and K be the p-Sylow. Note that H and K are normal subgroups of G. By counting element, |G|=|HK| and H交集K = {e}. So G = H x| K, the semidirect product. Now determine the homomorphism K → Aut(H) Since p does NOT divide q-1, this homomorphism must be identity. And this implies the semidirect product is in fact a direct product. Therefore, G is cyclic. (H, K are cyclic with orders p,q and (p,q)=1) : 3.o(G)=257，prove G is cyclic group(這題應該要找p-sylow subgroup嗎? : 這是找到了要怎麼證它是循環群呢?) : 先感謝指點迷津的大大 257 是質數... -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 219.71.210.134

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yusd24

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