# Why is the $p-$multiplicator of a $p-$group is an elementary abelian $p -$group?

Let $$G$$ be a $$p-$$group that have the finite presentation $$F/R$$($$F$$ is a free group of rank $$d$$); The $$p-$$multiplicator of $$G$$ is defined by $$G^* = R/[F,R]R^p$$ Why $$G^*$$ in an elementary abelian $$p -$$group?

Many thanks.

• This is directly from the way it is defined. You have $[R,R]\subseteq [F,R]$ so it is abelian, and you have included $R^p$ so it has exponent $p$. – Tobias Kildetoft Nov 30 '18 at 14:54