Centralizer product theorem

For faithful group actions
Suppose $$p,q$$ are distinct primes. Let $$P$$ be a finite $$p$$-group, and $$Q \le \operatorname{Aut}(P)$$ be an abelian $$q$$-group that is not cyclic. Suppose $$x_1, x_2, \dots, x_n$$ are the non-identity elements of $$Q$$, enumerated in any arbitrary order. Then, we have:

$$P = C_P(x_1)C_P(x_2) \dots C_P(x_n)$$.

For general group actions
Suppose $$p,q$$ are distinct primes. Let $$P$$ be a finite $$p$$-group, and $$Q$$ be an abelian $$q$$-group that is not cyclic. Suppose $$Q$$ acts on $$P$$ by automorphisms. Suppose $$x_1, x_2, \dots, x_n$$ are the non-identity elements of $$Q$$, enumerated in any arbitrary order. Then, we have:

$$P = C_P(x_1)C_P(x_2) \dots C_P(x_n)$$.

Note that the version for general group actions follows directly from the version for faithful group actions. In fact, for an action that is not faithful, it suffices to take the products of centralizers of coset representatives of the kernel of the action.

Particular cases

 * Centralizer product theorem for elementary abelian group

Corollaries/applications

 * Corollary of centralizer product theorem for rank at least three
 * Thompson transitivity theorem

Facts used

 * 1) uses::Centralizer product theorem for elementary abelian group
 * 2) uses::Omega-1 of center is normality-large in nilpotent p-group (in fact, socle equals Omega-1 of center in nilpotent p-group)
 * 3) uses::Central implies normal

Proof
Given: Primes $$p \ne q$$. A finite $$p$$-group $$P$$. An abelian non-cyclic subgroup $$Q \le \operatorname{Aut}(P)$$. $$x_1, x_2, \dots, x_n$$ are the non-identity elements of $$Q$$, enumerated in any order.

To prove: $$P = C_P(x_1)C_P(x_2) \dots C_P(x_n)$$.

Proof: We prove this by induction on the order of $$P$$, combined with fact (1). Fact (1) settles the case for $$P$$ an elementary abelian $$p$$-group.