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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. Edge Cover Through Edge Coloring

   https://doi.org/10.37236/13163  

   Chen, Guantao; Shan, Songling (April 2025, The Electronic Journal of Combinatorics)

   Let $$G$$ be a multigraph. A subset $$F$$ of $E(G)$ is an edge cover of $$G$$ if every vertex of $$G$$ is incident to an edge of $$F$$. The cover index, $$\xi(G)$$, is the largest number of edge covers into which the edges of $$G$$ can be partitioned. Clearly $$\xi(G) \le \delta(G)$$, the minimum degree of $$G$$. For $$U\subseteq V(G)$$, denote by $E^+(U)$ the set of edges incident to a vertex of $$U$$. When $|U|$ is odd, to cover all the vertices of $$U$$, any edge cover needs to contain at least $(|U|+1)/2$ edges from $E^+(U)$, indicating $$ \xi(G) \le |E^+(U)|/ ((|U|+1)/2)$$. Let $$\rho_c(G)$$, the co-density of $$G$$, be defined as the minimum of $|E^+(U)|/((|U|+1)/2)$ ranging over all $$U\subseteq V(G)$$, where $$|U| \ge 3$$ and $|U|$ is odd. Then $$\rho_c(G)$$ provides another upper bound on $$\xi(G)$$. Thus $$\xi(G) \le \min\{\delta(G), \lfloor \rho_c(G) \rfloor \}$$. For a lower bound on $$\xi(G)$$, in 1978, Gupta conjectured that $$\xi(G) \ge \min\{\delta(G)-1, \lfloor \rho_c(G) \rfloor \}$$. Gupta himself verified the conjecture for simple graphs, and Cao et al. recently verified this conjecture when $$\rho_c(G)$$ is not an integer. In this paper, we confirm the conjecture when the maximum multiplicity of $$G$$ is at most two or $$ \min\{\delta(G)-1, \lfloor \rho_c(G) \rfloor \} \le 6$$. The proof relies on a newly established result on edge colorings. The result holds independent interest and has the potential to significantly contribute towards resolving the conjecture entirely. 

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3. Khintchine-type recurrence for 3-point configurations

   https://doi.org/10.1017/fms.2022.97  

   Ackelsberg, Ethan; Bergelson, Vitaly; Shalom, Or (January 2022, Forum of Mathematics, Sigma)

   Abstract The goal of this paper is to generalise, refine and improve results on large intersections from [2, 8]. We show that if G is a countable discrete abelian group and $$\varphi , \psi : G \to G$$ are homomorphisms, such that at least two of the three subgroups $$\varphi (G)$$ , $$\psi (G)$$ and $$(\psi -\varphi )(G)$$ have finite index in G , then $$\{\varphi , \psi \}$$ has the large intersections property . That is, for any ergodic measure preserving system $$\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$$ , any $$A\in \mathcal {X}$$ and any $$\varepsilon>0$$ , the set $$ \begin{align*} \{g\in G : \mu(A\cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A)>\mu(A)^3-\varepsilon\} \end{align*} $$ is syndetic (Theorem 1.11). Moreover, in the special case where $$\varphi (g)=ag$$ and $$\psi (g)=bg$$ for $$a,b\in \mathbb {Z}$$ , we show that we only need one of the groups $aG$ , $bG$ or $(b-a)G$ to be of finite index in G (Theorem 1.13), and we show that the property fails, in general, if all three groups are of infinite index (Theorem 1.14). One particularly interesting case is where $$G=(\mathbb {Q}_{>0},\cdot )$$ and $$\varphi (g)=g$$ , $$\psi (g)=g^2$$ , which leads to a multiplicative version of the Khintchine-type recurrence result in [8]. We also completely characterise the pairs of homomorphisms $$\varphi ,\psi $$ that have the large intersections property when $$G = {{\mathbb Z}}^2$$ . The proofs of our main results rely on analysis of the structure of the universal characteristic factor for the multiple ergodic averages $$ \begin{align*} \frac{1}{|\Phi_N|} \sum_{g\in \Phi_N}T_{\varphi(g)}f_1\cdot T_{\psi(g)} f_2. \end{align*} $$ In the case where G is finitely generated, the characteristic factor for such averages is the Kronecker factor . In this paper, we study actions of groups that are not necessarily finitely generated, showing, in particular, that, by passing to an extension of $$\textbf {X}$$ , one can describe the characteristic factor in terms of the Conze–Lesigne factor and the $$\sigma $$ -algebras of $$\varphi (G)$$ and $$\psi (G)$$ invariant functions (Theorem 4.10). 

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4. 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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5. The Jordan–Chevalley decomposition for 𝐺-bundles on elliptic curves

   https://doi.org/10.1090/ert/631  

   Frăţilă, Dragoş; Gunningham, Sam; Li, Penghui (December 2022, Representation Theory of the American Mathematical Society)

   We study the moduli stack of degree 0 0 semistable G G -bundles on an irreducible curve E E of arithmetic genus 1 1 , where G G is a connected reductive group in arbitrary characteristic. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups H H of G G (the E E -pseudo-Levi subgroups), where each stratum is computed in terms of H H -bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan–Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where E E has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra g \mathfrak {g} (respectively, algebraic group G G ). We also provide a Tannakian description of these moduli stacks and use it to show that if E E is not a supersingular elliptic curve, the moduli of framed unipotent bundles on E E are equivariantly isomorphic to the unipotent cone in G G . Finally, we classify the E E -pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples. 

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