Suppose $G$ is a group. Is $G$ a subgroup of some group $H$ such that:

• $H$ is centerless;
• If $h \in H$ is an element of prime order, then $h \in G$.

In other words, can every group be embedded in a centerless group without introducing new prime order elements?

Suppose $G$ is a group. Is $G$ a subgroup of some group $H$ such that:

• $H$ is centerless;
• If $h \in H$ is an element of prime order $p$, then there is also some $g \in G$ of order $p$.

In other words, does any group embed in a centerless group without introducing an element of a new prime order?

This is a modified version of my question; the previous version turned out to be trivial.