Question

Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that (Q,+) is not cyclic.

Answer 1

Given, G is a cylic group.

Therefore, G=<a>.

And H is subgroup of G.

H=<a^i>.

Index,

$\dfrac{G}{H} ={a\hat{j}+<a\hat{i}>}$

If j>i then by division algorithm, j=ir+s

Then $a\hat{j}+<a\hat{i}>=a\hat{s}+<a\hat{i}>$

So, $\dfrac{G}{H}={a\hat{s}+<a\hat{i}>, 0<=s<i}$

$O(\dfrac{G}{H})$ = finite =index of H.

For( Q,+), choose H=$<\dfrac{1}{2}>$

Clearly $O(\dfrac{G}{H})=$ inifinte. 

So, G=(Q,+) is not cyclic.