# Structure of a single automorphism of a finite abelian p-group

A finite abelian $$p$$-group $$H$$ is homogenous when it is the direct sum of cyclic groups of the same order $$p^r$$, i.e. $$H \cong \big(\mathbb{Z}/p^{r}\mathbb{Z}\big)^{e}$$. Every finite abelian $$p$$-group $$G$$ can be decomposed into homogenous factors $$G = H_{d_1} \oplus H_{d_2} \oplus \cdots \oplus H_{d_k}$$ where $$H_{d_i}$$ has exponent $$p^{d_i}$$ and $$d_1 > d_2 > \cdots > d_k > 0$$. While this decomposition is unique up to isomorphism, unless $$k = 1$$ there are many distinct choices for the homogenous subgroups themselves.

Now suppose $$\varphi$$ is an automorphism of $$G$$ of order coprime to $$p$$. Must there be a homogenous decomposition that is invariant under $$\varphi$$?

If it helps, you can assume that the order of $$\varphi$$ is prime. For the intended application, all I really need is an invariant homogenous subgroup of the largest possible exponent.

## 1 Answer

It is true that if $$\phi$$ is automorphism of order prime to $$p$$ of a finite Abelian $$p$$-group $$A$$, then $$A$$ is a direct sum of indecomposable $$\phi$$-invariant subgroups ( that is, no summand is a direct sum of two non-zero $$\phi$$-invariant subgroups). It is also true that each of these indecomposable summands is homocyclic, that is,a direct sum of cyclic subgroups of the same order, ( what you call homogeneous). Hence the answer to your question is yes. This can be found in D.Gorenstein's book "Finite Groups" (the original results may be due to P. Hall, I am not sure).

• Thanks! The precise reference in Gorenstein is Theorem 5.2.2 on page 176. Commented Jun 19 at 20:01
• Glad you found the precise reference. Commented Jun 19 at 20:22
• I guess it's true for any action of a finite group of order coprime to $p$?
– YCor
Commented Jun 20 at 7:00
• @YCor: Indeed that is the case( and is done in Gorenstein). Commented Jun 20 at 7:34