The baby TKK algebra is a core of the extended affine Lie algebra of type A 1 over a semilattice in ޒ 2 . In this paper, we classify the irreducible integrable weight modules for the extended baby TKK algebra under the assumption that its center acts nontrivially.
Let G be a finite group. We say that (G, H, α) is a strongly stable triple if H ≤ G, α ∈ Irr(H) and (αG)H is a multiple of α. In this paper, we study the quasi-primitivity, inductors, and stabilizer limits of strongly stable triples. We show that under certain conditions all stabilizer limits of a strongly stable triple have equal degrees, thus generalizing the corresponding theorem of character triples due to Isaacs.
The normal embedding problem of finite solvable groups into [Formula: see text]-groups was studied. It was proved that for a finite solvable group [Formula: see text], if [Formula: see text] has a special normal nilpotent Hall subgroup, then [Formula: see text] cannot be a normal subgroup of any [Formula: see text]-group; on the other hand, if [Formula: see text] has a maximal normal subgroup which is an [Formula: see text]-group, then [Formula: see text] can occur as a normal subgroup of an [Formula: see text]-group under some suitable conditions. The results generalize the normal embedding theorem on solvable minimal non-[Formula: see text]-groups to arbitrary [Formula: see text]-groups due to van der Waall, and also cover the famous counterexample given by Dade and van der Waall independently to the Dornhoff’s conjecture which states that normal subgroups of arbitrary [Formula: see text]-groups must be [Formula: see text]-groups.
We give a natural generalization of the Gagola characters of finite groups from the aspect of Galois conjugation of characters, and then present some criteria for the existence of such generalized Gagola characters. We also introduce the Galois pairs to obtain more structure information of a finite group possessing a generalized Gagola character, which seems to be of independent interest and worth further study.Mathematics Subject Classification. 20C15.
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