The automorphism group for p-central p-groups

A p-group G is p-central if Gp � Z(G), and G is p2-abelian if (xy)p2 = xp2 yp2 for all x; y ε G. We prove that for G a finite p2-abelian p-central p-group, excluding certain cases, the order of G divides the order of Aut(G). © 2012 University of Isfahan

A note on automorphisms of finite p-groups

Let G be a finite non-cyclic p-group of order at least p^3. If G has an abelian maximal subgroup, or if G has an elementary abelian centre and is not strongly Frattinian, then the order of G divides the order the its automorphism group

Groups of p-deficiency one

In a previous paper, Button and I proved that all finitely presented groups of p-deficiency greater than one are p-large. Here I prove that groups with a finite presentation of p-deficiency one possess a finite index subgroup that surjects onto the integers. This implies that these groups do not have Kazhdan's property (T). Additionally, I prove that the aforementioned result of Button and myself implies a result of Lackenby

A pro-p group with infinite normal Hausdorff spectra

Using wreath products, we construct a finitely generated pro-p group G with infinite normal Hausdorff spectrum with respect to the p-power series. More precisely, we show that this normal Hausdorff spectrum contains an infinite interval; this settles a question of Shalev. Furthermore, we prove that the normal Hausdorff spectra of G with respect to other filtration series have a similar shape. In particular, our analysis applies to standard filtration series such as the Frattini series, the lower p-series and the modular dimension subgroup series. Lastly, we pin down the ordinary Hausdorff spectra of G with respect to the standard filtration series. The spectrum of G for the lower p-series displays surprising new features