(Q,+) cannot have a subgroup of index 2

Nov 2015
7
0
Ann Arbor
I want to prove the group (Q,+) cannot have a subgroup of index 2.
I assume H is the subgroup of index 2.And k1 k2 k3 are three rational number, satisfying k1 +k2 =k3 and they don't belong to H.
If we can show there always exist such three number, it is easy to show the union H and k1H is not Q.
But I got stuck here.
 

Deveno

MHF Hall of Honor
Mar 2011
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Tejas
Hint: show $2\Bbb Q = \Bbb Q$ (that is, the map $x \to 2x$ is surjective).
 
Nov 2015
7
0
Ann Arbor
It is easy to prove this, since if x is rational number, x/2 must be a rational number .
But what to do next?
 

Deveno

MHF Hall of Honor
Mar 2011
3,546
1,566
Tejas
If $\Bbb Q$ has a subgroup $H$ of index $2$ it follows that $\Bbb Q/H$ is of order $2$, that is, for every $a \not\in H$, that $2a = a+a \in H$. Thus $2\Bbb Q \subseteq H$.