Centralizer product theorem for elementary abelian group
Suppose are distinct primes. Suppose is an elementary abelian -group, and is an abelian -group that is not cyclic. Let be an enumeration of the non-identity elements of . Then, we have:
where denotes the set of fixed points in of the automorphism .
- Centralizer product theorem: The same result, except that is now an arbitrary -group, rather than an elementary abelian group.
Given: Primes . An elementary abelian -group , a non-cyclic abelian -subgroup of . are the non-identity elements of .
To prove: .
- By fact (1), viewing as a vector space over the field of elements and as a group acting on this vector space, we can decompose as a direct sum of irreducible -modules, say .
- Let be the pointwise stabilizer of in . Then, is cyclic: is an abelian group of automorphisms of the vector space , and is irreducible. This forces to be cyclic.
- Each is a nontrivial subgroup of : This follows from the previous step and the fact that isn't cyclic.
- For each , pick a non-identity element of . Then, contains : This follows from the definition of .
- : Each is contained in for some non-identity element . Thus, is the sum of all , ranging over the non-identity elements of .
Reverting to multiplicative notation yields the result.