7 research outputs found
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
Applications of p-deficiency and p-largeness
We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of
p-largeness to show that a group having a finite presentation with p-deficiency
greater than 1 is large, which implies that Schlage-Puchta's infinite finitely
generated p-groups are not finitely presented. We also show that for all primes
p at least 7, any group having a presentation of p-deficiency greater than 1 is
Golod-Shafarevich, and has a finite index subgroup which is Golod-Shafarevich
for the remaining primes. We also generalise a result of Grigorchuk on Coxeter
groups to odd primes.Comment: 23 page
Costs, complexity and size relationships in the UK university system
In 2 vols.Available from British Library Document Supply Centre-DSC:DXN020368 / BLDSC - British Library Document Supply CentreSIGLEGBUnited Kingdo
The Amit–Ashurst conjecture for finite metacyclic p-groups
Acknowledgements: We thank Ainhoa Iñiguez for her initial involvement in the project, and we thank the referee for their helpful comments. This collaboration was initiated at a Functor Categories for Groups meeting, which was supported by a London Mathematical Society Joint Research Group grant. The views expressed are those of the authors and do not reflect the official policy or position of the ARCYBER, the Department of the Army, the Department of Defense, or the US Government.Funder: Lund UniversityAbstractThe Amit conjecture about word maps on finite nilpotent groups has been shown to hold for certain classes of groups. The generalised Amit conjecture says that the probability of an element occurring in the image of a word map on a finite nilpotent group G is either 0, or at least 1/|G|. Noting the work of Ashurst, we name the generalised Amit conjecture the Amit–Ashurst conjecture and show that the Amit–Ashurst conjecture holds for finite p-groups with a cyclic maximal subgroup.
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p-Basilica Groups
We consider a generalisation of the Basilica group to all odd primes: the p-Basilica groups acting on the p-adic tree. We show that the p-Basilica groups have the p-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. This provides the first examples of weakly branch groups with such properties. In addition, the p-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the Hausdorff dimensions of the p-Basilica groups. Lastly, we show that the p-Basilica groups do not possess maximal subgroups of infinite index and that they have infinitely many non-normal maximal subgroups
p-Basilica Groups
We consider a generalisation of the Basilica group to all odd primes: the p-Basilica groups acting on the p-adic tree. We show that the p-Basilica groups have the p-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. This provides the first examples of weakly branch groups with such properties. In addition, the p-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the Hausdorff dimensions of the p-Basilica groups. Lastly, we show that the p-Basilica groups do not possess maximal subgroups of infinite index and that they have infinitely many non-normal maximal subgroups