Fundamental Theorem of Cyclic Groups

When proving that every subgroup of a cyclic group is cyclic.

Let $$G = \langle a \rangle$$ and suppose that $$H$$ is a subgroup of $$G$$ and assume that $$H \ne \{e\}$$.

The author begins with the claim that $$H$$ contains an element of the form $$a^{t}$$, where $$t$$ is positive.

To verify this claim, he says,

Since $$G = \langle a \rangle$$, every element of H has the form $$a^{t}$$; and when $$a^{t}$$ belongs to $$H$$ with $$t<0$$, then $$a^{-t}$$ belongs to $$H$$ also and -t is positive.

I see the first part clearly, but I am not able to see clearly what the second part which begin after ';' is saying.

• Isnt he using the fact that its inverse must be in the subgroup $H$ ? Apr 26 '20 at 9:57
• yes wow good. But why is he mentioning negative integers when he could have simply used the property of inverse in a subgroup, and completely altering the statement? Apr 26 '20 at 10:01
• I belive the answer below will help you with that , at first he doenst know what type of element is there he just knows that there is one, so he has to assume it could have a negative or positive exponent. Apr 26 '20 at 10:08

It's easy, suppose you choose an arbitrary element $$p$$ from the subgroup $$H$$. Then as the elements of $$H$$ are also the elements of the group $$G=$$ then p is of the form $$a^t$$ for some $$t\in \mathbb{Z}$$. If this $$t$$ is positive then we are done but if $$t\lt0$$ then H being the subgroup $$a^{-t} \in H$$ and $$-t\gt0$$. So in any case we have $$a^s \in H$$ for $$s\gt0$$.
Asserting that $$G=\langle a\rangle$$ means that$$G=\{a^t\mid t\in\Bbb Z\}.$$So, the author has to assume that an element of $$H$$ may be of the form $$a^t$$, with $$t<0$$.
• By that definition, if $a^t\in H$, $t$ may be reater than or smaller than $0$. But if $a^t\in H$ with $t<0$, then its inverse belongs to $H$ too. And the inverse of $a^t$ is $a^{-t}$. Now, since t«$t<0$, $-t>0$. So, this proves that for some $t\in\Bbb N$, $a^t\in H$. Apr 26 '20 at 13:29
The point is that WLOG we may assume any $$g\in H$$ is of the form $$g=h^t$$ for some $$t\in\Bbb Z^+$$, just by using the fact that when $$a$$ generates, so does $$a^{-1}$$.