## Find the amount of subgroups of order $3$ and $21$ in non-cyclic abelian group of order $63$

Find the amount of subgroups of order $$3$$ and $$21$$ in non-cyclic abelian group of order $$63$$.

In first case I found the amount of elements that have order $$3$$ – there are $$8$$ of them, in second case there are $$48$$ elements of order $$21$$. How do I connect these values with the amount of subgroups now?

## Prove the non-cyclic inequality

if a, b, c>0, then prove: $$\frac{a+b}{\sqrt{a^{2}+a b+b^{2}+b c}}+\frac{b+c}{\sqrt{b^{2}+b c+c^{2}+c a}}+\frac{c+a}{\sqrt{c^{2}+c a+a^{2}+a b}} \geq 2+\sqrt{\frac{a b+b c+c a}{a^{2}+b^{2}+c^{2}}}$$

## Noncyclic subgroups of multiplicative group of integers mod n

I want to find an $$n$$ such that $$\mathbb{Z}/n\mathbb{Z}^{\times}$$ has a noncyclic subgroup, and I’m struggling to think of an example of such a subgroup. How can I construct such a subgroup without thinking about the cyclic subgroups generated by the elements of $$\mathbb{Z}/n\mathbb{Z}^{\times}$$?