Re: [代數] 幾題期中考的問題..
※ 引述《paggei (XD)》之銘言：
: (1) Prove that a group that has only a finite number of subgroups
: must be finite.
: (2) 舉一個無窮乘法循環群的例子
: (3) G is group, H is a subgroup of G
: Prove that the number of all distinct right cosets of H in G is equal to
: the number of all distinct left cosets of H in C
: 即使考完了還是不會(炸)
: 想請問這幾題如何做呢@@
(1)Let G be a group a屬於G and let <a> be the cyclic subgroup generated by a.
Consider U <a> = G
a屬於G
since G has only finite number subgroups
and since the set of all cyclic subgroups is a subset of all subgroups,
we have there are only finite number cyclic subgroups
says <a_1>,<a_2>, <a_3>....<a_m>.
Then G = U <a_i> i從一到m
claim that for each i , <a_i> is finite.
If <a_i> is not finite,
then <a_i ^k> for k屬於N are cyclic subgroups of <a_i>,
which is infinite many,and are also subgroups of G
This contradicting to G has only finite number subgroups.
Hence,G is finite union of finite set is also finite.
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