GVZ-groups, Flat groups, and CM-Groups

Abstract

We show that a group is a GVZ-group if and only if it is a flat group. We show that the nilpotence class of a GVZ-group is bounded by the number of distinct degrees of irreducible characters. We also show that certain CM-groups can be characterized as GVZ-groups whose irreducible character values lie in the prime field.

Partial GVZ-groups

“…In Theorem B of [2], we show when G is a GVZ-group that the nilpotence class of G is bounded by |cd(G)|, where cd(G) is the set of degrees of the irreducible characters of G. We now obtain a similar result regarding the nilpotence class of N when every character in Irr(G | N ) has central type.…”

“…We now present the proof of Theorem D, which is an adaptation of the usual Taketa argument. Our proof is nearly identical to our proof of Theorem B of [2] with just a few subtle differences. When G is a nilpotent group, we write c(G) for the nilpotence class of G, and we set G 1 = G and G i+1 = [G i , G] for i ≥ 1 for the terms of the lower central series for G. Following [12] Proof.…”

“…Thus, ν has central type and since ν ∈ Irr(N | N ′ ) was chosen arbitrarily, N is a GVZ-group. Now we show conclusion (2). Assume that N is a GVZ-group.…”

“…Such groups are necessarily nilpotent (see [17]); thus, it is not difficult to see that G is a GVZ-group if and only if G is the direct product of GVZ groups of prime power orders for different primes. Moreover, we show in [2] that G is a GVZ-group if and only if cl G (g) = g[g, G] for each element g ∈ G.…”

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

“…We also study a special sub-class of GVZ p-groups called CM p−1 p-groups. A group G is called a CM n -group if every normal subgroup of G appears as the kernel of at most n irreducible characters of G (see [6]). In Theorem 5, we prove the following result for CM p−1 -groups.…”