Finite Order of Elements in Groups with Normal Subgroups

[jmjlt88]

Proposition: If every element of G/H has finite order, and every element of H has finite order, then every element of G has finite order.

Proof: Let G be a group with normal subgroup H. Suppose that every element of G/H has finite order and that every element of H has finite order. We wish to show that every element of G has finite order. Let x be any element in G. Now, Hx is in G/H and by our assumption has finite order. Let n be the order of Hx. Then (Hx)ⁿ=Hxⁿ=H. Since Hxⁿ=H, xⁿ ε H.* Now the element xⁿ is in H and therefore has finite order. Let m be the order of xⁿ. Thus x^nm=e**, which implies that x in G has finite order.

QED

*Ha=H iff a ε H.

**This does not mean that mn is the order of x. There are two possibe cases. The first case is that mn is the order of x. The other case is that mn is a multiple of the order of x. But, in either case we see that the order of x in G is finite.

_____________________________________________

Do I have the right idea? Also, if correct, should a remark like ** be included in the proof. This is how I did the proof in my notes. I usually do the ** remark as a comment to myself to justify a statement.

Thanks! =)

 

[micromass]

That's good!

A remark like ** is not really needed in the proof. However, it is an interesting remark and it's good that you noticed it. A nice question would be to exhibit an actual example where mn is not the order (if such an example exists).

 

[jmjlt88]

Thanks Micromass! :) I have a quick question for you. After a thoughout treatment of Pinter's book [which by the way has a massive amount of exercises of which I am trying to do everyone] and Lang's Linear Algebra, would one be ready for something like Artin? Again, thank you so much for all your help!

 

[micromass]

Thanks Micromass! :) I have a quick question for you. After a thoughout treatment of Pinter's book [which by the way has a massive amount of exercises of which I am trying to do everyone] and Lang's Linear Algebra, would one be ready for something like Artin? Again, thank you so much for all your help!

Without a doubt, you'll be ready. Artin is the perfect follow-up book for Pinter. And if you managed to get through Lang, then Artin should not really be much a problem.