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Hello. I thank for your ask, in advance.

Let $G$ be a finite group and $G$ has an element of order $n$ such that $\pi(n)=\pi(G)$ where $\pi(n)$ denote the set of prime divisors of $n$ and $\pi(G)$ denote the set of prime divisor of $|G|$. What can be said about the structure of the group? I know in nilpotent group there exist such element.

Hello. I thank for your answer, in advance.

Let $G$ be a finite group and $G$ has an element of order $n$ such that $\pi(n)=\pi(G)$ where $\pi(n)$ denote the set of prime divisors of $n$ and $\pi(G)$ denote the set of prime divisor of $|G|$. What can be said about the structure of the group? I know in nilpotent group there exist such element.

I generalize my question. What is the relation between the $|\pi(G)|$ and the maximum $|\pi(n)|$, where $n$ range in all order elements of a finite group $G$?