Free powered group for a set of primes

Definition

Suppose $\pi$ is a set of primes and $S$ is a generating set. The free $\pi$-powered group on $S$, denoted $F(S,\pi )$ with $S$ identified as a subset of this group, is the unique (up to isomorphism and with embedding of $S$) group satisfying the following:

1. This group is $\pi$-powered.
2. It has no proper subgroup that contains $S$ and is also $\pi$-powered.
3. Any set map from $S$ to a group $G$ extends to a unique group homomorphism from $F(S,\pi )$ to $G$.

Guarantee of existence

Further information: There exist free powered groups for any set of primes and any size of generating set