In this paper we study the oscillatory behavior of the even order nonlinear delay differential equation

(1)

where

(i) denotes the order of differentiation with respect to t. The delay terms τi σi are assumed to be real-valued, continuous, non-negative, non-decreasing and bounded by a common constant M on the half line (t0, + ∞ ) for some t0 ≧ 0.

For 1 ≦ p < ∞, μ real, let Lμ, p denote the collection of functions f, Lebesgue measurable on (0, ∞ ), and such that ‖ f ‖μP < ∞ , where

(1.1)

Also, if X and Y are Banach spaces, denote by [X, Y] the collection of bounded linear operators from X to Y; [X, X] denote by [X].

In this paper we give sufficient conditions for an HNN group to have the following two properties:

(1) any two finitely generated subgroups intersect in a finitely generated subgroup;

(2) every finitely generated subgroup containing a non-trivial subnormal subgroup has finite index.

The following result is a particular case of the main theorem.

This paper continues the analysis of Clifford theory for the case of a finite group G, K a normal subgroup of G and G/K abelian which was developed in [7]. In [7] the permutation actions of G/K on the characters of K and of (G/K)^ on the characters of G were studied in relation to their effects on induction and restriction of group characters.