抽象代数 西罗定理(sylow)

抽象代数 西罗定理(sylow)
关于西罗定理(Sylow)滴..有点小疑问一直没搞明白,希望达人给解释解释
如果一个群G,|G|=10, 那么该群则有两个西罗p子群,一个是2-Sylow,一个5-Sylow,又由Sylow第三定理知道,5-Sylow是唯一的,且为正规子群(normal group),而2-Sylow有可能为唯一,有可能有5个. 5-sylow中有4个元素的order为5,2-sylow中有一个元素的order为2.(实在不知道order的中文是啥....lol).
我的问题是:对于群G,其元素的order可能为1,2,5,10, 在5-Sylow,除去identity,剩下四个元素是否是G中所有order5的元素?
2.若2-sylow不是正规子群,那么这个5个2-sylow的并集中除去identity的4个order为2的元素是否是G中所有order为2的元素?
望达人解答....(翻译成中文好累啊....估计有些词不达意,,贴上英文,)
Hi Dr McShane
I have a question about Sylow group.
If a group G and |G| = 10, then G has two Sylow subgroups which are 2-Sylow of order 2 and five Sylow of order 5. And by Sylow 3rd thm, the 5-sylow is normal subgroup of G. 2-sylow may be the normal or may have 5 of them.
my question is, for the group G it may has elements of order 1, 2, 5 and 10, and in the 5-Sylow subgoup it contians
4 elements of order 5(except identity), are those 4 element all the elements of order 5 in group G?
or may I put it in this way: Is it true that all the elements of order k contained in G will be contained in the union of all the corresponding
p-Sylow subgroup? Or this is only true for all elements of order P(prime) in G will be contained in the P-Sylow subgroup(here the P-Sylow is normal)?
再补充一点
如果|G|=20的话,有一个order4的2-sylow,那其中就含有order2与order4的元素.那么在G中order2的元素有几个?order4的呢?
一楼的问题如果该2-sylow仅有一个呢?