# On the automorphisms of a finite abelian group

I read the following statement in an article:

Let $$G$$ be a finite abelian group. Then $$G$$ can be decomposed into a direct product of abelian $$p$$-groups i.e. $$G\cong G_1 \times G_2 \times ... \times G_k$$. From here it follows that $$\operatorname{Aut}(G)\cong \operatorname{Aut}(G_1) \times \operatorname{Aut}(G_2)\times ... \operatorname{Aut}(G_k)$$.

There are a couple of things which I do not understand. The fact that $$G$$ can be decomposed into a direct product of abelian $$p$$-groups seems all right to me from the classification theorem of finite abelian groups. But the next part regarding the automorphism group seems to assume that the orders of $$G_1, G_2,... G_k$$ are all relatively prime. I don't understand why this is the case.
EDIT: After the answers I received I think I managed to understand how this works. These $$G_i$$'s are just the Sylow subgroups as someone pointed out and their orders are obviously relatively prime. I believe that this is used in proving the classification theorem of finite abelian groups, but we may as well just use the part I quoted in order to get a nice form for $$\operatorname{Aut}(G)$$.

• The groups $G_i$ are simply the Sylow subgroups of $G$. – the_fox Mar 2 at 13:24
• @the_fox thank you ! This proved to be really useful to me. I added my conclusion in the question. – user69503 Mar 2 at 14:49

The orders of the $$G_i$$ are indeed relatively prime. This comes from the theorem, whose proof you may want to go over, and the definition of $$p$$-group.

One of many sources: http://ramanujan.math.trinity.edu/bmiceli/teach/algebra1S14/AbelianGroups.pdf

The order of $$G_i$$ are relatively prime, but not the orders of $${\rm Aut}(G_i)$$.

For a given cyclic group $$G\cong C_n$$ we have $${\rm Aut}(C_n)\cong (\Bbb Z/n\Bbb Z)^{\times}$$ of order $$\phi(n)$$, see here:

Order of automorphism group of cyclic group

Hence for the group $$C_{p^k}$$ the automorphism group has order $$\phi(p^k)=p^k-p^{k-1}$$. These orders are not coprime in general for different primes $$p$$ and $$q$$.

In general, the automorphism group is given as follows:

The automorphism group of a direct product of abelian groups is isomorphic to a matrix group