Omega-1 of abelian p-group is coprime automorphism-faithful

Statement
Suppose $$P$$ is an Abelian group of prime power order. Let $$\Omega_1(P)$$ denote the subgroup generated by elements of order $$p$$ in $$P$$. Then, $$\Omega_1(P)$$ is a coprime automorphism-faithful subgroup of $$P$$: any non-identity automorphism of $$P$$ of order coprime to $$p$$ restricts to a non-identity automorphism of $$\Omega_1(P)$$.

Related facts

 * Omega-1 of odd-order p-group is coprime automorphism-faithful: Drops the assumption of Abelianness, but requires odd order.
 * Automorphism of finite 2-group of Mersenne prime order acts nontrivially on Omega-1

Textbook references

 * , Page 178, Theorem 2.4, Section 5.2 ($$p'$$-automorphisms of Abelian $$p$$-groups)