Search

Let 퐺 be a finite group. Recall that an 퐴-group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the diameter of the commuting graph of a solvable 퐴-group. Assuming that the commuting graph is connected, we show when the derived length of 퐺 is 2, the diameter of the commuting graph will be at most 4. In the general case, we show that the diameter of the commuting graph will be at most 6. In both cases, examples are provided to show that the upper bound of the commuting graph cannot be improved. [ABSTRACT FROM AUTHOR]

Let 𝐺 be a nonabelian group. We say that 𝐺 has an abelian partition if there exists a partition of 𝐺 into commuting subsets A1, A2 , . . . , An of 𝐺 such that |Ai| ⩾ 2 for each i = 1, 2, . . . , n. This paper investigates problems relating to groups with abelian partitions. Among other results, we show that every finite group is isomorphic to a subgroup of a group with an abelian partition and also isomorphic to a subgroup of a group with no abelian partition. We also find bounds for the minimum number of partitions for several families of groups which admit abelian partitions – with exact calculations in some cases. Finally, we examine how the size of a partition with the minimum number of parts behaves with respect to the direct product. [ABSTRACT FROM AUTHOR]

Let G be a solvable group, and let Δ(G) be the character degree graph of G. In this paper, we generalize the definition of a square graph to graphs that are block squares. We show that if G is a solvable group so that Δ(G) is a block square, then G has at most two normal nonabelian Sylow subgroups. Furthermore, we show that when G is a solvable group that has two normal nonabelian Sylow subgroups and Δ(G) is block square, then G is a direct product of subgroups having disconnected character degree graphs. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A conjugacy class of G is said to be vanishing if there exists an irreducible character of G which takes the value 0 on the elements of this class. In this note, we describe the groups whose vanishing classes all have size p for some prime p. [ABSTRACT FROM AUTHOR]

Let G be a group and n a positive integer. We define Bn(G) to be the intersection of the normalizers of all the non-n-subnormal subgroups of G. We give a new characterization for nilpotent groups in terms of a series defined via Bn(G). [ABSTRACT FROM AUTHOR]

Let G be a Camina p-group of nilpotence class 3. We prove that if G ′ < CG(G′), then |G3| ≤ |G′ : G3|1/2. We also prove that if G/G3 has only one or two abelian subgroups of order |G : G′|, then G′ < CG(G′). If G/G3 has pa + 1 abelian subgroups of order |G : G′|, then either G′ < CG(G′) or |Z(G) | ≤ p2a. [ABSTRACT FROM AUTHOR]

Let p be a prime. We show that other than a few exceptions, alternating groups will have p-blocks with small defect for p equal to 2 or 3. Using this result, we prove that a finite group G has a normal Sylow p-subgroup P and G/P is nilpotent if and only if φ.1/2 divides |G W ker.φ/| for every irreducible Brauer character φ of G. [ABSTRACT FROM AUTHOR]

If χ is an irreducible character of a finite group G, then the codegree of χ is ∣G: ker(χ)∣/χ(1). We show that if G is a p-group, then the nilpotence class of G is bounded in terms of the largest codegree for an irreducible character of G. [ABSTRACT FROM AUTHOR]