On primitive representations of soluble groups of finite rank

Abstract

Within the framework of the thermally activated process of the flux line or flux line bundles, and by time integration of the 1D equation of motion of the circulating current density J (ρ, t), which is suitable for thin superconducting films (R d, λ), we present numerical calculations of the current profiles, magnetization hysteresis loops and ac susceptibility χ n = χ n + iχ n for n = 1, 3 and 5 of a thin disc immersed in an axial time-dependent external magnetic field B a (t) = B dc + B ac cos(2πνt). Our calculated results are compared with those of the critical state model (CSM) and found to prove the approximate validity of the CSM below the irreversibility field. The differences between our computed results and those of the CSM are also discussed.

On the Primitive Irreducible Representations of Finitely Generated Linear Groups of Finite Rank

“…But then (17) implies that P / ∈ µ kA (akL/akL) and the obtained contradiction shows that µ kA (akL/akL) ⊆ µ kA (bkL/bkLI) If P ∈ µ kA (bkL/bkLI) and P / ∈ µ kA (akL/akL) then it follows from (17) that there is Q ∈ µ kA (akL/akLI) such that Q < P and (19) implies that Q ∈ µ kA (ckL/ckLI). But then (18) implies that P / ∈ µ kA (bkL/bkL) and the obtained contradiction shows that µ kA (akL/akL) ⊇ µ kA (bkL/bkLI). Thus, we can conclude that µ kA (akL/akL) = µ kA (bkL/bkLI).…”

“…The subgroup H ≤ G which provides (1) is said to be a control subgroup for the RG-module M. Various types of control subgroups for modules over group rings of groups of finite rank were considered in [15][16][17][18].…”

“…Naturally, the following question appears: what can be said on the construction of a group G if it has a faithful primitive irreducible representation over a field k? It should be noted that there are many results which show that the existence of a faithful irreducible representation of a group G over a field k may have essential influence on the structure of the group G (see for instance [11][12][13][14][18][19][20]).…”

“…It is well known that any polycyclic group is finitely generated soluble of finite rank and meets the maximal condition for subgroups (in particular, for normal subgroups). In [18] we showed that in the class of soluble groups of finite rank with the maximal condition for normal subgroups only polycyclic groups may have faithful irreducible primitive representations over a field of characteristic zero.…”

On the primitive representations of finitely generated metabelian groups of finite rank over a field of non-zero characteristic

“…It is well known (see [14]) that any polycyclic group is finitely generated soluble of finite rank and meets the maximal condition for subgroups (in particular, for normal subgroups). In [10] we showed that in the class of soluble groups of finite rank with the maximal condition for normal subgroups only polycyclic groups may have faithful irreducible primitive representations over a field of characteristic zero. In [7][8][9] we studied irreducible primitive representations of metabelian groups of finite rank over a field of characteristic zero.…”