Let $G$ a finite group. If $a\in G$, a consequence of the Lagrange's Theorem is that the order of $a$ divides the order of $G$. Let $p=|G|$. If $p$ is prime, it is well known that $G$ is cyclic, and therefore, there exists $a\in G$ such that the order of $a$ is $p$ and, of course, exist $b$ in $G$ such that the order of $b$ is $1$. If $n$ is a composite number, namely $$n=p_1^{m_1}p_2^{m_2}\ldots p_n^{m_n},$$ where $p_i$ are primes, my question is this:
Is there, for each $i$, an element $a_i$ in $G$ such that the order of $a_i$ is $p_i$? Is there, for each $i$, $b_i\in G$, such that the order of $b_i$ is $p_i^{m_i}$?