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1. The spread of a finite group

   https://doi.org/doi.org/10.4007/annals.2021.193.2.5  

   Burness, Tim; Guralnick, Robert; Harper, Scott (January 2021, Annals of mathematics)

   null (Ed.)

   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. 

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2. Rigid automorphisms of linking systems

   https://doi.org/10.1017/fms.2021.17  

   Glauberman, George; Lynd, Justin (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $$2$$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G , then it is of $p'$ -order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$ -subgroups. We present two applications of this last result, one to tame fusion systems. 

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3. Sparse graph counting and Kelley-Meka bounds for binary systems

   Filmus, Yuval; Hatami, Hamed; Hosseini, Kaave; Kelman, Esty (October 2024, 65th IEEE Symposium on Foundations of Computer Science (FOCS))

   In a recent breakthrough, Kelley and Meka (FOCS 2023) obtained a strong upper bound on the density of sets of integers without non-trivial three-term arithmetic progressions. In this work, we extend their result, establishing similar bounds for all linear patterns defined by binary systems of linear forms, where “binary” indicates that every linear form depends on exactly two variables. Prior to our work, no strong bounds were known for such systems even in the finite field model setting. A key ingredient in our proof is a graph counting lemma. The classical graph counting lemma, developed by Thomason (Random Graphs 1985) and Chung, Graham, and Wilson (Combinatorica 1989), is a fundamental tool in combinatorics. For a fixed graph H, it states that the number of copies of H in a pseudorandom graph G is similar to the number of copies of H in a purely random graph with the same edge density as G. However, this lemma is only non-trivial when G is a dense graph. In this work, we prove a graph counting lemma that is also effective when G is sparse. Moreover, our lemma is well-suited for density increment arguments in additive number theory. As an immediate application, we obtain a strong bound for the Turán problem in abelian Cayley sum graphs: let Γ be a finite abelian group with odd order. If a Cayley sum graph on Γ does not contain any r-elique as a sub graph, it must have at most 2−Ωr(log1/16|Γ|)⋅|Γ|2 edges. These results hinge on the technology developed by Kelley and Meka and the follow-up work by Kelley, Lovett, and Meka (STOC 2024). 

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4. 𝐺-cohomologically rigid local systems are integral

   https://doi.org/10.1090/tran/8610  

   Klevdal, Christian; Patrikis, Stefan (February 2022, Transactions of the American Mathematical Society)

   Let G G be a reductive group, and let X X be a smooth quasi-projective complex variety. We prove that any G G -irreducible, G G -cohomologically rigid local system on X X with finite order abelianization and quasi-unipotent local monodromies is integral. This generalizes work of Esnault and Groechenig [Selecta Math. (N. S. ) 24 (2018), pp. 4279–4292; Acta Math. 225 (2020), pp. 103–158] when G = G L n G= \mathrm {GL}_n , and it answers positively a conjecture of Simpson [Inst. Hautes Études Sci. Publ. Math. 75 (1992), pp. 5–95; Inst. Hautes Études Sci. Publ. Math. 80 (1994), pp. 5–79] for G G -cohomologically rigid local systems. Along the way we show that the connected component of the Zariski-closure of the monodromy group of any such local system is semisimple; this moreover holds when we relax cohomological rigidity to rigidity. 

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5. THE CYCLIC GRAPH OF A Z -GROUP

   https://doi.org/10.1017/S0004972720001318  

   COSTANZO, DAVID G.; LEWIS, MARK L.; SCHMIDT, STEFANO; TSEGAYE, EYOB; UDELL, GABE (January 2020, Bulletin of the Australian Mathematical Society)

   null (Ed.)

   Abstract: For a group G, we define a graph Delta (G) by letting G^#=G\{1} be the set of vertices and by drawing an edge between distinct elements x,y in G^# if and only if the subgroup is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate Delta (G) for a Z-group G. 

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