Abstract We consider a generalisation of the Basilica group to all odd primes: p -Basilica groups acting on -adic tree. show that have -congruence subgroup property but not congruence nor weak property. This provides first examples weakly branch with such properties. In addition, give branch, which are super strongly fractal. compute orders quotients these groups, enable us determine Hausdorff ...

For simplicity, we use the following convention: G is a group, a, b are elements of G, m, n are natural numbers, and p is a prime natural number. One can prove the following propositions: (1) If for every natural number r holds n 6= pr, then there exists an element s of N such that s is prime and s | n and s 6= p. (2) For all natural numbers n, m such that n | pm there exists a natural number r...

In the 1970s, Isaacs conjectured that there should be a logarithmic bound for the length of solvability of a p-group G with respect to the number of different irreducible character degrees of G. So far, there are just a few partial results for this conjecture. In this note, we say that a pro-p group G has property (I) if there is a real number D = D(G) that just depends on G such that for any o...

Let $p$ be a prime. A pro-$p$ group $G$ is said to 1-smooth if it can endowed with homomorphism of groups $G\to1+p\mathbb{Z}_p$ satisfying formal version Hilbert 90. By Kummer theory, maximal Galois fields containing root 1 order $p$, together the cyclotomic character, are 1-smooth. We prove that finitely generated $p$-adic analytic if, and only occurs as field $p$. This gives positive answer D...