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Let $G$ be an abelian (not elementary) finite $p$-group. In what conditions the automorphism group of $G$ is solvable?

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No, this is false. Consider the centralizer of an element of the form $I_n+E_{1,n}$ in $GL_n(p)$, with $p$ prime and $n\geq 5$. It is an extension of a non-abelian $p$-group. – Dima Pasechnik Dec 29 '12 at 17:15
If you're interested in automorphism groups of p-groups more generally, take a look at: MR2320459 (2008h:20035) Reviewed Helleloid, Geir T.(1-STF); Martin, Ursula(4-LNDQM-C) The automorphism group of a finite p-group is almost always a p-group. (English summary) J. Algebra 312 (2007), no. 1, 294–329. 20D45 (20D15) – Dan Ramras Dec 29 '12 at 17:25
up vote 2 down vote accepted

I believe that for $p\ge5$, these are the direct products of cyclic groups of pairwise distinct orders, for if $C_{p^e}\times C_{p^e}$ is a direct factor of $G$, then $GL(2,p)$ is a homomorphic image of a subgroup of $A=\text{Aut}(G)$.

Similarly for $p=2$ or $3$, I expect the groups $G$ you are looking for are those where no order in the direct product appears more than $2$ times.

In order to show that these groups indeed have a solvable automorphism group, you probably may apply induction: Let $N$ be the subgroup of $G$ generated by the $p$-th powers in $G$. Then $N$ has the same shape as above (with the order of each direct factor divided by $p$). So $\text{Aut}(N)$ is solvable by induction. As $C_A(N)$ is the kernel of the restriction of $A$ to $N$, all what remain to show is that $C_A(N)$ is solvable too.

Alternatively, one could try to use the induction hypothesis for $G/K$, where $K$ is generated by the elements of order $p$. Note that $N$ is (non-canonically) isomorphic to $G/K$.

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Thanks so much for your nice answer. – majid arezoomand Jan 14 '13 at 6:38

The Frattini quotient G/Phi(G) is the maximal elementary abelian quotient of G; the group of automorphisms of G which act trivially on G/Phi(G) is a p-group, and so Aut(G) is solvable if and only if its image in the linear group Aut(G/Phi(G)) is; that's at least a partial answer. Also, this holds for general pro-p groups, not only for finite abelian p-groups.

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Thanks for your attention. Your answer is a well-know fact, Indeed I want a classification of abelian $p$-groups with solvable automorphism. – majid arezoomand Dec 29 '12 at 19:36

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