Find smallest subgroups

Question

I'm struggling with this. Let $G$ be an infinite cyclic group and let $g$ be its generator ($G=\langle g \rangle$). Find the smallest subgroup that contains $g^4$ and $g^6$. By definition it is $\langle\{g^4,g^6\}\rangle$ but the problem requires me to give a generator for this subgroup I would appreciate any advice

Answer 1

Note that the generator of any subgroup of the cyclic group $\langle g\rangle$ must be a power of $g$, say $g^k$. Now for some $m$, $g^m\in\langle g^k\rangle$ implies that $m$ is a multiples of $k$. Hence the smallest subgroup of $\langle g\rangle$ that contains both $g^4$ and $g^6$ must be $\langle g^2\rangle$ as $2=\gcd(4,6)$.

Answer 2

Hint: So let $g^x$ be in that, then you should be able to write it as $$g^{4\cdot a+6\cdot b}=g^{2(2\cdot a+3\cdot b)}.$$ Can you show that every single number that you can imagine can be written as $2\cdot a+3\cdot b$? If it helps, notice that $2\cdot 2-3\cdot 1=1.$

Answer 3

The smallest subgroups containing $g^4$ and $g^6$ is the subgroup generated by these elements, and, the group being commutative, it is equal to the set of elements $$(g^4)^r(g^6)^s=g^{4r+6s}\quad (r,s\in\mathbf Z).$$ Now the ideal of $\mathbf Z$ generated by two elements is the ideal generated by their g.c.d., so $$\langle\, g^4,g^6\rangle=\langle\, g^2\rangle.$$