Minimal degrees of faithful quasi-permutation representations for direct products of p-groups

On faithful quasi-permutation representations of $VZ$-groups and Camina $p$-groups

“…Let χ, ψ ∈ X such that χ ∈ Irr(G|Z(G)) and ψ ∈ nl(G/Z(G)). Then ker χ < Z(G) ≤ ker ψ ⇒ ker χ ∩ ker ψ = ker χ and this contradict equation (11). Therefore, X must be a subset of Irr(G|Z(G)).…”

“…Since G has exactly two character degrees, all the Sylow subgroups of G, except one, are abelian. = c(G) (see [12]).…”

“…Let χ be an arbitrary irreducible character in Irr(G|Z(G)). Then χ ↓ Z(G) = χ(1)λ, where λ ∈ lin(Z(G)), and (11) ker χ = ker λ < Z(G) < G ′ , ∀ χ ∈ Irr(G|Z(G)).…”

“…Hence X ⊆ nl(G) = Irr(G|Z(G)) ⊔ nl(G/Z(G)) (disjoint union). Further, since X must satisfy equation (11), X can't be a subset of nl(G/Z(G)). Now, suppose X intersect with Irr(G|Z(G)) and nl(G/Z(G)).…”

“…Study of c(G), q(G) and µ(G) have been done by several reseachers in past. The interested reader may refer to [1,2,3,4,5,7,11,12,25,26]. It is easy to see that for a finite group G, we have the following relation between µ(G), c(G), q(G) :…”