Give an example..

**ynj** · September 5th 2009, 05:57 PM

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

**NonCommAlg** · September 5th 2009, 06:52 PM

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 $\mathbb{Z}(p^{\infty}).$

**ynj** · September 5th 2009, 08:33 PM

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 $H=\left \{\frac{m}{p^{2n}} + \mathbb{Z}: \ \ m, n \in \mathbb{Z}, \ n \geq 0 \right \} \subset G?$

**NonCommAlg** · September 5th 2009, 09:10 PM

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 $H=G.$

**ynj** · September 5th 2009, 09:32 PM

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 !

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