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A group G is said to be 3/2-generated if every nontrivial element belongs to a generating pair. It is easy to see that if G has this property, then every proper quotient of G is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if G is a finite group and every proper quotient of G is cyclic, then for any pair of nontrivial elements x1, x2 ϵ G, there exists y ϵ G such that G = ⟨x1, y⟩ = ⟨x2, y⟩. In other words, s(G) ⩾ 2, where s(G) is the spread of G. Moreover, if u(G) denotes the more restrictive uniform spread of G, then we can completely characterise the finite groups G with u(G) = 0 and u(G) = 1. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods, and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups with socle an exceptional group of Lie type, and this is the case we treat in this paper. 

Macrophages are essential for skeletal muscle homeostasis, but how their dysregulation contributes to the development of fibrosis in muscle disease remains unclear. Here, we used single-cell transcriptomics to determine the molecular attributes of dystrophic and healthy muscle macrophages. We identified six clusters and unexpectedly found that none corresponded to traditional definitions of M1 or M2 macrophages. Rather, the predominant macrophage signature in dystrophic muscle was characterized by high expression of fibrotic factors, galectin-3 (gal-3) and osteopontin (Spp1). Spatial transcriptomics, computational inferences of intercellular communication, and in vitro assays indicated that macrophage-derived Spp1 regulates stromal progenitor differentiation. Gal-3⁺macrophages were chronically activated in dystrophic muscle, and adoptive transfer assays showed that the gal-3⁺phenotype was the dominant molecular program induced within the dystrophic milieu. Gal-3⁺macrophages were also elevated in multiple human myopathies. These studies advance our understanding of macrophages in muscular dystrophy by defining their transcriptional programs and revealSpp1as a major regulator of macrophage and stromal progenitor interactions.