Principal congruence subgroups of the Hecke groups and related resultsIkikardes, Sebahattin; Sahin, Recep; Naci Cangul, I.
doi: 10.1007/s00574-009-0023-ypmid: N/A
In this paper, first, we determine the quotient groups of the Hecke groups H(λ
q
), where q ≥ 7 is prime, by their principal congruence subgroups H
p
(λ
q
) oflevel p, where p is also prime. We deal with the case of q = 7 separately, because of its close relation with the Hurwitz groups. Then, using the obtained results, we find the principal congruence subgroups of the extended Hecke groups
$$
\overline H
$$
(λ
q
) for q ≥ 5 prime. Finally, we show that some of the quotient groups of the Hecke group H(λ
q
) and the extended Hecke group
$$
\overline H
$$
(λ
q
), q ≥ 5 prime, by their principal congruence subgroups H
p
(λ
q
) are M*-groups.
The influence of M-supplemented subgroups on the structure of finite groupsMiao, Long
doi: 10.1007/s00574-009-0024-xpmid: N/A
A subgroup H of a group G is said to be M-supplemented in G if there exists a subgroup B of G such that G = HB and T B < G for every maximal sub-group T of H. Moreover, a subgroup H is called c-supplemented in G if there exists a subgroup K such that G = HK and H ∩ K ≤ H
G
where H
G
is the largest normal subgroup of G contained in H. In this paper we give some conditions of supersolv-ability of finite group under assumption that some primary subgroups have some kinds of supplements, which are generalizations of some recent results.
Families of periodic orbits in resonant reversible systemsLima, Maurício; Teixeira, Marco
doi: 10.1007/s00574-009-0025-9pmid: N/A
We study the dynamics near an equilibrium point p
0 of a Z
2(ℝ)-reversible vector field in ℝ2n
with reversing symmetry R satisfying R
2 = I and dimFix(R) = n. We deal with one-parameter families of such systems X
λ such that X
0 presents at p
0 a degenerate resonance of type 0: p: q. We are assuming that the linearized system of X
0 (at p
0) has as eigenvalues: λ1 = 0 and λ
j
= ±iα
j
, j = 2, … n. Our main concern is to find conditions for the existence of one-parameter families of periodic orbits near the equilibrium.
On ramification in the compositum of function fieldsAnbar, Nurdagül; Stichtenoth, Henning; Tutdere, Seher
doi: 10.1007/s00574-009-0026-8pmid: N/A
The aim of this paper is twofold: Firstly, we generalize well-known formulas for ramification and different exponents in cyclic extensions of function fields over a field K (due to H. Hasse) to extensions E = F(y), where y satisfies an equation of the form f(y) = u · g(y) with polynomials f(y), g(y) ∈ K[y] and u ∈ F. This result depends essentially on Abhyankar’s Lemma which gives information about ramification in a compositum E = E
1
E
2 of finite extensions E
1, E
2 over a function field F. Abhyankar’s Lemma does not hold if both extensions E
1/F and E
2/F are wildly ramified. Our second objective is a generalization of Abhyankar’s Lemma if E
1/F and E
2/F are cyclic extensions of degree p = char(K). This result may be useful for the study of wild towers of function fields over finite fields.
On calibrated and separating sub-actionsGaribaldi, Eduardo; Lopes, Artur; Thieullen, Philippe
doi: 10.1007/s00574-009-0028-6pmid: N/A
We consider a one-sided transitive subshift of finite type σ: Σ → Σ and a Hölder observable A. In the ergodic optimization model, one is interested in properties of A-minimizing probability measures. If Ā denotes the minimizing ergodic value of A, a sub-action u for A is by definition a continuous function such that A ≥ u ○ σ − u + Ā. We call contact locus of u with respect to A the subset of Σ where A = u ○ σ − u + Ā. A calibrated sub-action u gives the possibility to construct, for any point x ε Σ, backward orbits in the contact locus of u. In the opposite direction, a separating sub-action gives the smallest contact locus of A, that we call Ω(A), the set of non-wandering points with respect to A.