- #1
jmjlt88
- 96
- 0
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 xnm=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.
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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! =)
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 xnm=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! =)