Structure lemma for p-group with coprime automorphism group having automorphism trivial on invariant subgroups

Statement
Suppose $$G$$ is a fact about::group of prime power order, and $$A$$ is a subgroup of $$\operatorname{Aut}(G)$$, of order relatively prime to $$G$$. Suppose there exists a non-identity element $$\varphi \in A$$ such that $$\varphi$$ acts as the identity on every proper $$A$$-invariant subgroup of $$G$$. Then, we have the following conclusions about the structure of $$G$$:


 * 1) $$[G,G] = \Phi(G) \le Z(G)$$. In other words, the commutator subgroup is contained in the center, and the quotient by the commutator subgroup is elementary Abelian.
 * 2) Either of two cases can occur: $$\Phi(G) = Z(G)$$, in which case $$G$$ is a fact about::special group, or $$\Phi(G)$$ is trivial, in which case $$G$$ is an fact about::elementary Abelian group.

We also have the following conclusions about the action of $$A$$ on $$G$$:


 * 1) $$\varphi$$ acts nontrivially on $$G/G'$$, and the action of $$A$$ on $$G/G'$$ is irreducible (in other words, there are no proper nontrivial $$A$$-invariant subgroups).
 * 2) In case $$G$$ is a special group, $$\varphi$$ also acts trivially on $$G'$$.

Facts used

 * 1) uses::Stability group of subnormal series of p-group is p-group
 * 2) uses::Omega-1 of Abelian p-group is coprime automorphism-faithful
 * 3) uses::Maschke's averaging lemma

Proof
Given: A $$p$$-group $$G$$, a subgroup $$A$$ of $$\operatorname{Aut}(G)$$ such that $$A$$ has order relatively prime to $$G$$. A non-identity element $$\varphi \in A$$ such that $$\varphi$$ acts as the identity on every proper $$A$$-invariant subgroup.

To prove: $$\varphi$$ acts nontrivially on $$G/G'$$ and trivially on $$G'$$.$$G/G'$$ is elementary Abelian and $$A$$ acts irreducibly on $$G/G'$$. $$G$$ is either elementary Abelian or special.

Proof that $$\varphi$$ acts nontrivially on $$G/G'$$
Since $$G'$$ is a characteristic subgroup of $$G$$, it is in particular $$A$$-invariant. Thus, $$\varphi$$ acts trivially on $$G'$$. If $$\varphi$$ acted trivially on $$G/G'$$, fact (1) would yield that $$\varphi$$ is the identity map, a contradiction. Thus, $$\varphi$$ acts nontrivially on $$G/G'$$. In particular, $$A$$ acts nontrivially on $$G/G'$$.

Proof that $$G/G'$$ is elementary Abelian and that the action of $$A$$ is irreducible
Let $$\overline{G} = G/G'$$. Suppose the action of $$A$$ on $$G$$ is decomposable, so $$\overline{G} = \overline{G_1} \times \overline{G_2}$$, where both $$\overline{G_1}$$ and $$\overline{G_2}$$ are proper $$A$$-invariant. Their inverse images in $$G$$ are proper $$A$$-invariant subgroups $$G_1, G_2 \le G$$, and $$\varphi$$ acts trivially on both $$G_1$$ and $$G_2$$. Thus, $$\varphi$$ acts trivially on $$G_1G_2 = G$$, a contradiction.

Thus, the action of $$A$$ on $$\overline{G}$$ is indecomposable.

Next, consider the induced action of $$A$$ on $$\Omega_1(\overline{G})$$. Suppose $$\Omega_1(\overline{G})$$ is a proper subgroup of $$\overline{G}$$. Then, the inverse image of this in $$G$$ is a proper $$A$$-invariant subgroup, so $$\varphi$$ acts trivially on it, forcing $$\varphi$$ to act trivially on $$\Omega_1(\overline{G})$$. By fact (2), we see that this forces $$\varphi$$ to be the identity on $$\overline{G}$$, a contradiction.

Thus, $$\Omega_1(\overline{G}) = \overline{G}$$, so $$\overline{G}$$ is elementary Abelian. The action of $$A$$ on $$\overline{G}$$ is indecomposable, so by fact (3), it is irreducible.

Proof that $$G$$ is elementary Abelian or special
We first show that $$G'$$ centralizes $$G$$.

Thus, $$G' \le Z(G)$$. So, $$Z(G)$$ is an $$A$$-invariant subgroup containing $$G'$$. Thus, $$Z(G)/G'$$ is an $$A$$-invariant subgroup of $$\overline{G}$$. By the irreducibility of the action of $$A$$ on $$\overline{G}$$, either $$Z(G)/G'$$ is trivial or $$Z(G)/G' = \overline{G}$$. Thus, either $$Z(G) = G$$ (in which case $$G'$$ is trivial and $$G$$ is elementary Abelian) or $$Z(G) = G'$$ (in which case $$G' = Z(G) = \Phi(G)$$, and so $$G$$ is a special group).

Textbook references

 * , Theorem 3.7, Page 181-183, Section 5.3 ($$p'$$-automorphisms of $$p$$-groups)