Centralizer product theorem for elementary abelian group

Statement

Suppose $p,q$ are distinct primes. Suppose $P$ is an elementary abelian $p$-group, and $Q\leq \operatorname {Aut} (P)$ is an abelian $q$-group that is not cyclic. Let $x_{1},x_{2},\dots ,x_{n}$ be an enumeration of the non-identity elements of $Q$. Then, we have:

$P=C_{P}(x_{1})C_{P}(x_{2})\dots C_{P}(x_{n})$.

where $C_{P}(x_{i})$ denotes the set of fixed points in $P$ of the automorphism $x_{i}$.

Related facts

Generalizations

• Centralizer product theorem: The same result, except that $P$ is now an arbitrary $p$-group, rather than an elementary abelian group.

Facts used

1. Maschke's averaging lemma

Proof

Given: Primes $p\neq q$. An elementary abelian $p$-group $P$, a non-cyclic abelian $q$-subgroup $Q$ of $\operatorname {Aut} (P)$. $x_{1},x_{2},\dots ,x_{n}$ are the non-identity elements of $Q$.

To prove: $P=C_{P}(x_{1})C_{P}(x_{2})\dots C_{P}(x_{n})$.

Proof:

1. By fact (1), viewing $P$ as a vector space over the field of $p$ elements and $Q$ as a group acting on this vector space, we can decompose $P$ as a direct sum of irreducible $Q$-modules, say $P_{1},P_{2},\dots ,P_{r}$.
2. Let $Q_{i}$ be the pointwise stabilizer of $P_{i}$ in $Q$. Then, $Q/Q_{i}$ is cyclic: $Q/Q_{i}$ is an abelian group of automorphisms of the vector space $P_{i}$, and $P_{i}$ is irreducible. This forces $Q/Q_{i}$ to be cyclic.
3. Each $Q_{i}$ is a nontrivial subgroup of $Q$: This follows from the previous step and the fact that $Q$ isn't cyclic.
4. For each $i$, pick a non-identity element $x$ of $Q_{i}$. Then, $C_{P}(x)$ contains $P_{i}$: This follows from the definition of $Q_{i}$.
5. $P=\sum C_{P}(x_{i})$: Each $P_{i}$ is contained in $C_{P}(x)$ for some non-identity element $x$. Thus, $P$ is the sum of all $C_{P}(x)$, $x$ ranging over the non-identity elements of $P$.

Reverting to multiplicative notation yields the result.

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 69, Theorem 3.3, Chapter 3, Section 3 (Complete reducibility), ^{More info}