On M-groups with Sylow towers

Connecting monomiality questions with the structure of rational group algebras

“…Hence ψ ′ is linear and N ′ / ker ψ ′ is abelian, using Lemma 2.27 of [14]. Thus, G ′ / ker ψ ′ is abelian-by-supersolvable, due to Lemma 2.2 of [4]. Since abelian-by-supersolvable groups are generalized strongly monomial, we have that χ ′ is generalized strongly monomial.…”

“…Furthermore, a triple τ ′ is said to be a multilinear reduction of τ if there is a series of triples τ = τ 0 , τ 1 , • • • , τ n = τ ′ such that each τ i is a linear reduction of τ i−1 for 1 ≤ i ≤ n, and τ ′ is called a linear limit if it is a multilinear reduction of τ in a way that the only possible linear reduction of τ ′ is τ ′ itself. Also, we recollect some notions introduced in [4] which are related to the work in [5,17,18]. For a triple τ = (G, N, ψ) with N G and ψ ∈ Irr(N), the section of τ , denoted Secτ , is defined to be the quotient group N/Z(ψ G ).…”

“…Observe that the existence of a successful search of (G, N, ψ), in Theorem 7.4 of [20], implies that it has a trivial linear limit. Following the notation of Chang, Zheng and Jin [4], we write N ∝ G to denote those normal subgroups N of G for which the triple (G, N, ψ) has a trivial linear limit for each ψ ∈ Irr(N). Hence Parks's result [20] can be rephrased as follows: if N is a nilpotent normal subgroup of a monomial group G such that the factor group G/N is supersolvable of odd order, then N ∝ G.…”

“…Later, generalizing the ideas contained in Parks's and Loukaki's work, the theory of linear limits was introduced by Dade and Loukaki [5]. Using this theory, Chang, Zheng and Jin [4] generalized the work of Gunter [9] for a class of solvable groups with technical conditions involved in it. Extending the work of Parks [20], the monomiality of normal subgroups and Hall subgroups of a class of solvable groups with some restrictions on its supersolvable residual was given by Zheng and Jin [23].…”

“…In sections 4-7, we examined all the groups where monomiality questions have been answered and surprisingly found that all of them are indeed generalized strongly monomial. To be precise, we have shown in series of theorems (Theorems 2-10) that all the classes of groups discussed in the work of Parks [20], Gunter [9], Loukaki [18], Lewis [16], Chang, Zheng and Jin [4] and Zheng and Jin [23] turn out to be generalized strongly monomial. Below is the list of some well known classes of groups which have been proved to be generalized strongly monomial:…”