Example for application of theorem: $G = \langle C_G(a) \mid a \in Q \setminus \{1\}\rangle$

Question

Let $p, q$ be distinct prime numbers, $G$ a $p$-group, and $Q$ a non-cyclic abelian $q$-group of automorphisms of $G$. Then, $$ G = \langle C_G(a) \mid a \in Q \setminus \{1\} \rangle . $$

Hey guys,

In our lecture, we proved the theorem mentioned above, and our professor provided an example where we can apply it. However, the example was quite strange and complex. I'm wondering if anyone knows of a group where this lemma can be used more straightforwardly. I've been experimenting a bit myself but haven't been able to come up with a good idea yet.

So far, I've considered a semidirect product involving a $2 \neq p$-group and the Klein four-group. This setup would give me a non-cyclic abelian q-group (where $ q \neq p $) acting on a $p$-group. However, I haven't found a suitable group on which the Klein four-group operates. Can anyone help me out here or suggest a better example? Thanks a lot in advance!

Answer 1

Here is a possible example. Take $p=3, q=2$. Consider the group $$ \left\{ \begin{pmatrix} \epsilon_1 & a & b\\ 0& \epsilon_2 &c\\ 0 & 0 & \epsilon_3 \end{pmatrix} : \epsilon_1, \epsilon_2,\epsilon_3, a,b,c\in\mathbb{F}_3, \text{ and }\epsilon_1 \epsilon_2 \epsilon_3 =1 \right\} $$ which is a semidirect product of $$ G=\left\{ \begin{pmatrix} 1 & a & b\\ 0& 1 &c\\ 0 & 0 & 1 \end{pmatrix} : a,b,c\in\mathbb{F}_3 \right\} $$ of order $27=p^3$, by $$ Q=\left\{ \begin{pmatrix} \epsilon_1 & 0 & 0\\ 0& \epsilon_2 &0\\ 0 & 0 & \epsilon_3 \end{pmatrix} : \epsilon_1, \epsilon_2,\epsilon_3\in\mathbb{F}_3, \text{ and }\epsilon_1 \epsilon_2 \epsilon_3 =1 \right\} $$ which is noncyclic of order $4=q^2$.