LetN be a square free integer, and let A be an optimal elliptic curve over Q of conductor N . We prove that if A has a rational torsion point of prime order r such that r does not divide 6N , then r divides the order of the cuspidal subgroup of J0(N).

Let p ≥ 5 be a prime number. We prove that the abstract commensurator of the Nottingham group N (Fp) coincides with its automorphism group, which is known to be a finite extension of N (Fp). As a corollary we deduce that the Nottingham group cannot be embedded as an open subgroup of a topologically simple group.

We give a general approach to N -periodic sequences over a finite field Fq constructed via a subgroup D of the group of invertible elements modulo N . Well known examples are Legendre sequences or the two-prime generator. For some generalizations of sequences considered in the literature and for some new examples of sequence constructions we determine the linear complexity.

Let p > 3 be a prime, ζp a pth primitive root of 1, and ∆ the Galois group of Q(ζp) over Q. Let q 6= p be a prime and n the order of q modulo p. Assume q 6≡ 1 mod p and so n ≥ 2, p(q− 1)|qn − 1, and n|p− 1. Set f = (q − 1)/p and e = (p− 1)/n. Let Q be a prime ideal of Z[ζp] above q and let F = Z[ζp]/Q. Thus F ∼= Fqn , the finite field with q elements. Let α ∈ Z[ζp] be a generator of F such that...

the theorem 12 in [a note on $p$-nilpotence and solvability of finite groups, j. algebra 321(2009) 1555--1560.] investigated the non-abelian simple groups in which some maximal subgroups have primes indice. in this note we show that this result can be applied to prove that the finite groups in which every non-nilpotent maximal subgroup has prime index aresolvable.

Let $$a\ge 1$$ and $$n>1$$ be odd integers. For a given prime p, we prove under certain conditions that the class groups of imaginary quadratic fields $${\mathbb {Q}}(\sqrt{a^2-4p^n})$$ have subgroup isomorphic to {Z}}/n{\mathbb {Z}}$$ . We also show this family has infinitely many members with property their In addition, deduce some unconditional results concerning divisibility numbers fields....

Let G be a group and H a subgroup. It is shown that the set of indices {[H:HngHg~] | geG) has a finite upper bound n if and only if there is a normal subgroup N < G which is commensurable with //; i.e., such that [H:Nr\H] and [N:Nr>H] are finite; moreover, the latter indices admit bounds depending only on n. If the bounded index hypothesis is assumed only for g running over some subgroup K<G, t...