Cyclic Groups!
Hey there, fellow math lovers. :)
I am in an intro group theory course for my undergrad (I'm a music ed major, math minor) and I have a couple problems here that I could use your guidance on. Browsing through previous posts, I see a few other LJers also in a similar course. Superb.
I have 4 questions:
1: I had to list all the cyclic subgroups of S3 (think the symmatries of a triangle) - which I can do, but then I need to explain whether S3 has a noncyclic proper subgroup. I don't think so, but I'm not quite sure why.
2: I need to show some examples of finite (cyclic) subgroups of C* (the complex numbers except 0). But, never having worked with complex numbers before, the only one I can come up with is ={1,-1,i,-i}. Can someone help me think of another one or two, or at least how I can come up with one?
3: So given H and K cyclic subgroups of some Abelian group G with the order of H=10, and order of K=14, I have to show that G contains a cyclic subgroup of order 70. Now, intuitively, yeah G should have a subgroup of order 10*7=70. But I'm not quite sure how to show it. If I knew G was cyclic, then I could easily say 10 and 14 must both divide the order of G (by a theorem) and so then |G|=10n and =14m for some n,m integers. And then somehow I would manipulate this to get that 70 divides G too. Buuuuut, since I don't know G is cyclic, I only know it's Abelian, I don't know how do do this. Are all Abelian groups cyclic? I'm confused.
4: If G is a group with no nontrivial proper subgroups, then I need to show that G must be cyclic. Then what does this tell me about the order of G. I'm not quite sure what to do here. First I would assume that G itself was not the trivial group and have some a in G where a is not e and then consider as a subgroup of G. But I'm not quite sure where to go.
I know that seems like I'm majorly confused (and I probably am), but apart from those 4 questions I think I have an ok understanding of what is going on so far.
Hints/Help much appreciated. Thank you. :)
I am in an intro group theory course for my undergrad (I'm a music ed major, math minor) and I have a couple problems here that I could use your guidance on. Browsing through previous posts, I see a few other LJers also in a similar course. Superb.
I have 4 questions:
1: I had to list all the cyclic subgroups of S3 (think the symmatries of a triangle) - which I can do, but then I need to explain whether S3 has a noncyclic proper subgroup. I don't think so, but I'm not quite sure why.
2: I need to show some examples of finite (cyclic) subgroups of C* (the complex numbers except 0). But, never having worked with complex numbers before, the only one I can come up with is ={1,-1,i,-i}. Can someone help me think of another one or two, or at least how I can come up with one?
3: So given H and K cyclic subgroups of some Abelian group G with the order of H=10, and order of K=14, I have to show that G contains a cyclic subgroup of order 70. Now, intuitively, yeah G should have a subgroup of order 10*7=70. But I'm not quite sure how to show it. If I knew G was cyclic, then I could easily say 10 and 14 must both divide the order of G (by a theorem) and so then |G|=10n and =14m for some n,m integers. And then somehow I would manipulate this to get that 70 divides G too. Buuuuut, since I don't know G is cyclic, I only know it's Abelian, I don't know how do do this. Are all Abelian groups cyclic? I'm confused.
4: If G is a group with no nontrivial proper subgroups, then I need to show that G must be cyclic. Then what does this tell me about the order of G. I'm not quite sure what to do here. First I would assume that G itself was not the trivial group and have some a in G where a is not e and then consider as a subgroup of G. But I'm not quite sure where to go.
I know that seems like I'm majorly confused (and I probably am), but apart from those 4 questions I think I have an ok understanding of what is going on so far.
Hints/Help much appreciated. Thank you. :)