Is every element of a group an element of a Sylow p-subgroup of that group?

Question

Just to clarify, I'm thinking of finite abelian groups, but it may be true in general.

This feels like it's obvious, but I'm not quite sure why. Maybe someone can justify it?

(Without using that the group is isomorphic to the product of its Sylow p-subgroups, since I feel that there's a more simple reason.)

Answer 1

Certainly not.

An element of a Sylow $p$-subgroup as or a power of $p$. Hence any element that is not of prime power order is not in a Sylow subgroup. Example: A generator of the cyclic group of order $6$.