CEU Electronic Theses and Dissertations, 2016
| Author | Aslan, Tugba |
|---|---|
| Title | On the structure of large powers and random generation in the Nottingham group |
| Summary | The Nottingham group, N, is the group of formal power series over a finite field where the group operation is substitution. In the theory of the Nottingham group, power and commutator structures play a significant role. We confirm a conjecture that was posed by K. Keating in [Kea05] for most of its cases, but we also show that this conjecture is not true for the remaining few cases. In more details, we give the sharp upper bounds for the distance between large p-th powers of any given two elements of N which share same depth and some leading coefficients. The key idea of our work is to extend some matrix that was introduced by Keating in [Kea05]. With this extended matrix, we could prove the sharpness of the proposed upper bounds and, moreover, we could show that the depths of large powers of two elements which satisfy Keating’s bound grow as slow as possible. Keating’s matrix can provide a very useful tool in the study of the Nottingham group. In this connection, we use this matrix to tackle a conjecture that was posed by A. Shalev “Any two random elements of the Nottingham group generate an open subgroup with probability 1”. We could confirm this conjecture, but for the elements obeying some extra restrictions. We also argue some possible improvements of our approach that might be helpful for confirming Shalev’s conjecture completely. |
| Supervisor | Hegedus, Pal |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2016/aslan_tugba.pdf |
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