Geodesic
A criterion of elementary equivalence of automorphism groups of reduced Abelian $p$-groups
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 5, pp. 157-163

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Consider reduced Abelian $p$-groups ($p\geq3$) $A_1$ and $A_2$. In this paper, we prove that the automorphism groups $\operatorname{Aut}A_1$ and $\operatorname{Aut}A_2$ are elementary equivalent if and only if the groups $A_1$ and $A_2$ are equivalent in second-order logic bounded by the cardinalities of the basic subgroups of $A_1$ and $A_2$. 
@article{FPM_2012_17_5_a9,
     author = {M. A. Roizner},
     title = {A criterion of elementary equivalence of automorphism groups of reduced {Abelian} $p$-groups},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {157--163},
     publisher = {mathdoc},
     volume = {17},
     number = {5},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2012_17_5_a9/}
}
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M. A. Roizner. A criterion of elementary equivalence of automorphism groups of reduced Abelian $p$-groups. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 5, pp. 157-163. http://geodesic.mathdoc.fr/item/FPM_2012_17_5_a9/