Answer to Question #188089 in Abstract Algebra for Renu

Question

prove that every non trival subgroup of a cyclic group has finite index hence prove that (Q,+) is non cyclic

Answer 1

Solution:

Given, G is a cyclic group.

Therefore, G=<a>.

And H is subgroup of G.

H=<$a^i$ >.

Index,

G/H={$a^j+<a^i>$ }

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

Then $a^j+<a^i>=a^s+<a^i>$

So, G/H={$a^s+<a^i>, 0<=s<i$ }

O(G/H)= finite =index of H.

For( Q,+), choose H=<1/2>

Clearly o(G/H)= inifinte. 

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