Burnside's theorem on coprime automorphisms and Frattini subgroup

Statement
Suppose $$P$$ is a finite $$p$$-group, i.e., a group of prime power order. Suppose $$\varphi$$ is an automorphism of $$P$$ whose order (as an element of the automorphism group of $$P$$) is relatively prime to $$P$$. Then, if $$\varphi$$ induces the identity on $$P/\Phi(P)$$, $$\varphi$$ is the identity automorphism of $$P$$. Here $$\Phi(P)$$ is the Frattini subgroup of $$P$$.

Equivalently the kernel of the map:

$$\operatorname{Aut}(P) \to \operatorname{Aut}(P/\Phi(P))$$

is a $$p$$-group. We also say that the Frattini subgroup of a $$p$$-group is a quotient-coprime automorphism-faithful subgroup.

Generalizations

 * Frattini subgroup of finite group is quotient-coprime automorphism-faithful: This generalizes the statement of Burnside's theorem to an arbitrary finite group.

Similar facts

 * Stability group of subnormal series of p-group is p-group: The result is similar, and it uses a similar proof technique.
 * Omega-1 of odd-order p-group is coprime automorphism-faithful

Facts used

 * 1) uses::Frattini subgroup of finite group is quotient-coprime automorphism-faithful

Proof
Given: A finite $$p$$-group $$P$$, an automorphism $$\varphi$$ of $$P$$ whose order is relatively prime to $$p$$. Further $$\varphi$$ induces the identity automorphism on $$P/\Phi(P)$$.

To prove: $$\varphi$$ is the identity automorphism.

Proof: The proof follows directly from fact (1), which is actually the general formulation for finite groups.

(NOTE: We can use the Burnside's basis theorem instead; instead of picking a coset representative for every coset, we pick a Burnside basis, and then find a coset representative for each coset in the Burnside basis, that is fixed under $$\sigma$$.)

Textbook references

 * , Page 199, Theorem 1.4 (Chapter 5)
 * , Page 199, Exercise 26(e) and (f) (Section 6.1)