1
$\begingroup$

Question:

Determine the subgroup lattice for $\mathbb{Z}_{p^2 q}$ where $p, q$ are distinct primes.

Indeed, $\left | \mathbb{Z}_{p^2 q} \right | = p^2 q$

The unique subgroups of $\mathbb{Z}_{p^2 q}$ are of the form $\left \langle \frac{p^2 q}{k} \right \rangle$ for each positive divisor $k$ of $n$.

My knowledge of primes are lacking. Any hint(s) would be helpful. Maybe $k$ divides $p^2 q$ implies $k$ divides $p^2$ or $k$ divides $q$ might be helpful?

Thanks in advance.

  • 1
    HINT: you can for a divisor of $p^2q$ only by multiplying $p$ and $q$. As a consequence, $$\{ 1,p,q,p^2,pq,p^2q\}$$ is the whole set of divisors of $p^2q$ – 2017-02-28
  • 0
    Just a little TeX note: I suppose it's good form to always put your exponents in braces, but when the exponent is a single letter or digit, it's not strictly necessary. As Trump works to kill net neutrality, it will become more important to save bandwidth by cutting out that which is not strictly necessary. – 2017-02-28

0 Answers 0