Lewis, Mark Lanning; Lytkina, Daria V.; Mazurov, Viktor Danilovich; Moghaddamfar, Ali Reza Splitting via noncommutativity. (English) Zbl 1404.20017 Taiwanese J. Math. 22, No. 5, 1051-1082 (2018). Summary: Let \(G\) be a nonabelian group and \(n\) a natural number. We say that \(G\) has a strict \(n\)-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup \(A\) and \(n\) nonempty subsets \(B_1, B_2, \ldots, B_n\), such that \(|B_i| > 1\) for each \(i\) and within each set \(B_i\), no two distinct elements commute. We show that every finite nonabelian group has a strict \(n\)-split decomposition for some \(n\). We classify all finite groups \(G\), up to isomorphism, which have a strict \(n\)-split decomposition for \(n = 1,2,3\). Finally, we show that for a nonabelian group \(G\) having a strict \(n\)-split decomposition, the index \(|G:A|\) is bounded by some function of \(n\). Cited in 1 Document MSC: 20D25 Special subgroups (Frattini, Fitting, etc.) 20E34 General structure theorems for groups 20D05 Finite simple groups and their classification Keywords:strict \(n\)-split decomposition; simple group; commuting graph PDFBibTeX XMLCite \textit{M. L. Lewis} et al., Taiwanese J. Math. 22, No. 5, 1051--1082 (2018; Zbl 1404.20017) Full Text: DOI arXiv Euclid References: [1] M. Akbari and A. R. Moghaddamfar, The existence or nonexistence of non-commuting graphs with particular properties, J. Algebra Appl. 13 (2014), no. 1, 1350064, 11 pp. · Zbl 1292.20025 [2] ——–, Groups for which the noncommuting graph is a split graph, Int. J. Group Theory 6 (2017), no. 1, 29–35. [3] G. Higman, Suzuki \(2\)-groups, Illinois J. Math. 7 (1963), 79–96. · Zbl 0112.02107 [4] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967. · Zbl 0217.07201 [5] B. Huppert and N. Blackburn, Finite Groups II, Springer-Verlag, Berlin, 1982. · Zbl 0477.20001 [6] ——–, Finite Groups III, Springer-Verlag, Berlin, 1982. [7] I. M. Isaacs, Finite Group Theory, Graduate Studies in Mathematics 92, American Mathematical Society, Providence, RI, 2008. · Zbl 1169.20001 [8] M. Suzuki, A new type of simple groups of finite order, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 868–870. · Zbl 0093.02301 [9] ——–, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105–145. · Zbl 0106.24702 [10] E. P. Vdovin, Maximal orders of abelian subgroups in finite simple groups, Algebra and Logic 38 (1999), no. 2, 67–83. · Zbl 0936.20009 [11] V. I. Zenkov, Intersections of abelian subgroups in finite groups, Math. Notes 56 (1994), no. 1-2, 869–871. · Zbl 0839.20032 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.