1. Give an example..

An infinite group has no proper infinite subgroup. Give an example.

2. Originally Posted by ynj
An infinite group has no proper infinite subgroup. Give an example.
our very own Prufer group: fix a prime number $p$ and define $G=\left \{\frac{m}{p^n} + \mathbb{Z}: \ \ m, n \in \mathbb{Z}, \ n \geq 0 \right \} \subset \mathbb{Q}/\mathbb{Z}.$ every proper subgroup of $G$ is finite and cyclic. in fact $H_n=<1/p^n + \mathbb{Z}>,$ where the

integer $n \geq 0,$ is fixed, are exactly all proper subgroups of $G.$ this is easy to prove and i'll leave it for you. a standard notation for $G$ is $\mathbb{Z}(p^{\infty}).$

3. Originally Posted by NonCommAlg
our very own Prufer group: fix a prime number $p$ and define $G=\left \{\frac{m}{p^n} + \mathbb{Z}: \ \ m, n \in \mathbb{Z}, \ n \geq 0 \right \} \subset \mathbb{Q}/\mathbb{Z}.$ every proper subgroup of $G$ is finite and cyclic. in fact $H_n=<1/p^n + \mathbb{Z}>,$ where the

integer $n \geq 0,$ is fixed, are exactly all proper subgroups of $G.$ this is easy to prove and i'll leave it for you. a standard notation for $G$ is $\mathbb{Z}(p^{\infty}).$
How about $H=\left \{\frac{m}{p^{2n}} + \mathbb{Z}: \ \ m, n \in \mathbb{Z}, \ n \geq 0 \right \} \subset G?$

4. Originally Posted by ynj
How about $H=\left \{\frac{m}{p^{2n}} + \mathbb{Z}: \ \ m, n \in \mathbb{Z}, \ n \geq 0 \right \} \subset G?$
this is not a "proper" subgroup because then $\frac{m}{p^{2n-1}} + \mathbb{Z} =\frac{mp}{p^{2n}} + \mathbb{Z} \in H,$ for all integers $m,n$ with $n \geq 1,$ and so $H=G.$

5. Originally Posted by NonCommAlg
this is not a "proper" subgroup because then $\frac{m}{p^{2n-1}} + \mathbb{Z} =\frac{mp}{p^{2n}} + \mathbb{Z} \in H,$ for all integers $m,n$ with $n \geq 1,$ and so $H=G.$
hmm..I have got it... If $\frac{p^km}{p^n}$appears, where $\gcd (m,p)=1$, then $\frac{1}{p^{k}},k\leq n-m$appears.. If there is infinite elements in a subgroup, then $n-m\leq M$for every element. And this will be a contradiction...
Great! I have thought of your example before, but I was stuck on my example..Now I get it !