Journal of Group Theory

Abstract A proper subgroup 𝐻 of a finite group 𝐺 is called large if | H | 3 ≥ | G | \lvert H\rvert^{3}\geq\lvert G\rvert . This paper characterises all the large maximal subgroups of almost simple groups whose socle is a finite simple classical group. For such a socle G 0 G_{0} and a core-free subgroup H 0 H_{0} , define Q ⁢ ( G 0 , H 0 ) \mathcal{Q}(G_{0},H_{0}) as the set of almost simple groups 𝐺 with socle G 0 G_{0} that have a large maximal subgroup 𝐻 satisfying H 0 = H ∩ G 0 H_{0}=H\cap G_{0} . We establish necessary and sufficient conditions on the pair ( G 0 , H 0 ) (G_{0},H_{0}) for the set Q ⁢ ( G 0 , H 0 ) \mathcal{Q}(G_{0},H_{0}) to be nonempty.

Abstract We prove that the pro-(finite supersolvable) closure of a finitely generated subgroup of the free group is finitely generated. This extends similar results for pro-𝑝 closures proved by Ribes–Zalesskii and pro-(finite nilpotent) closures proved by Margolis–Sapir–Weil.

Abstract In this article, we study rational representations of groups of order p 5 p^{5} , where 𝑝 is an odd prime. For a 𝑝-group 𝐺 and an irreducible complex character 𝜒 of 𝐺, the construction of an irreducible rational matrix representation of 𝐺 affording the character Ω ⁢ ( χ ) \Omega(\chi) is equivalent to determining a pair ( H , ψ ) (H,\psi) , with 𝐻 a subgroup of 𝐺 and 𝜓 a linear character of 𝐻 such that ψ G = χ \psi^{G}=\chi and Q ⁢ ( ψ ) = Q ⁢ ( χ ) \mathbb{Q}(\psi)=\mathbb{Q}(\chi) , where Ω ⁢ ( χ ) = m Q ⁢ ( χ ) ⁢ ∑ σ ∈ Gal ⁢ ( Q ⁢ ( χ ) / Q ) χ σ \Omega(\chi)=m_{\mathbb{Q}}(\chi)\sum_{\sigma\in\mathrm{Gal}(\mathbb{Q}(\chi)/\mathbb{Q})}\chi^{\sigma} and m Q ⁢ ( χ ) m_{\mathbb{Q}}(\chi) denotes the Schur index of 𝜒 over ℚ. For each inequivalent irreducible rational representation of every group of order p 5 p^{5} , we determine such a pair, referred to as a required pair. We also derive combinatorial formulations for the Wedderburn decomposition of rational group algebras of these 𝑝-groups, using results from their rational representations.

For an irreducible complex character 𝜒 of a finite group 𝐺, the codegree of 𝜒 is defined by | G : ker ( χ ) | / χ ( 1 ) \lvert G:\ker(\chi)\rvert/\chi(1) , where ker ⁡ ( χ ) \ker(\chi) is the kernel of 𝜒. In this paper, we give a detailed characterization of finite groups of non-prime-power order with exactly four (irreducible) character codegrees.

Recently, the author published a book [A. A. Ivanov, Ever-Evolving Groups—an Introduction to Modern Finite Group Theory , Algebr. Appl. 32, Springer, Cham, 2025] where he summarised the recent progress in the geometric theory of sporadic groups and outlined some geometries which require further investigation. Among them was a geometry for the smallest Conway sporadic simple group Co 3 \mathrm{Co}_{3} , with diagram originally introduced in [M. A. Ronan and G. Stroth, Minimal parabolic geometries for the sporadic groups, European J. Combin. 5 (1984), 1, 59–91] (cf. [M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs , London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316–331, Table 1] and [F. Buekenhout, Diagram geometries for sporadic groups, Finite Groups—Coming of Age (Montreal 1982), Contemp. Math. 45, American Mathematical Society, Providence (1985), geometry (23), p. 14]), which we denote by G ⁢ ( Co 3 ) \mathcal{G}(\mathrm{Co}_{3}) . The 2-local geometries for the other Conway groups Co 1 \mathrm{Co}_{1} and Co 2 \mathrm{Co}_{2} are the tilde and Petersen geometries which have been intensively studied (cf. [A. A. Ivanov and S. V. Shpectorov, The flag-transitive tilde and Petersen-type geometries are all known, Bull. Amer. Math. Soc. (N. S.) 31 (1994), 2, 173–184]). However, G ⁢ ( Co 3 ) \mathcal{G}(\mathrm{Co}_{3}) seems to be studied less. There is another geometry associated with Co 3 \mathrm{Co}_{3} (cf. [M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs , London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316–331, Table 1] and [F. Buekenhout, Diagram geometries for sporadic groups, Finite Groups—Coming of Age (Montreal 1982), Contemp. Math. 45, American Mathematical Society, Providence (1985), geometry (23), p. 14]) with diagram The elements of type 1, 2, 3, and 4 in the geometry G 276 \mathcal{G}_{276} correspond to cliques of size 1, 2, 3, and 6, respectively, in a double cover of the complete graph on 276 vertices. The group Co 3 \mathrm{Co}_{3} acts doubly transitively on the vertex set of the complete graph and flag-transitively on G 276 \mathcal{G}_{276} . This double cover is naturally associated with the well-known 2-graph of Co 3 \mathrm{Co}_{3} . The geometry G 276 \mathcal{G}_{276} is simply connected, as established by [M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs , London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316–331, Proposition 6]. In the present paper, we recover the distance 2 graph of G 276 \mathcal{G}_{276} from the universal cover of G ⁢ ( Co 3 ) \mathcal{G}(\mathrm{Co}_{3}) , in particular re-proving the simple connectedness of the latter geometry originally established in [A. Chermak, B. Oliver and S. Shpectorov, The linking systems of the Solomon 2-local finite groups are simply connected, Proc. Lond. Math. Soc. (3) 97 (2008), 1, 209–238]. A key step in our proof makes use of the simple connectedness of the Petersen-type geometry of the McLaughlin group, as shown in [B. Baumeister, A. A. Ivanov and D. V. Pasechnik, A characterization of the Petersen-type geometry of the McLaughlin group, Math. Proc. Cambridge Philos. Soc. 128 (2000), 1, 21–44].

Given a finite group 𝐺 and a conjugacy class of involutions 𝑋 of 𝐺, we define the commuting involution graph C ⁢ ( G , X ) \mathcal{C}(G,X) to be the graph with vertex set 𝑋 and x , y ∈ X x,y\in X adjacent if and only if x ≠ y x\neq y and x ⁢ y = y ⁢ x xy=yx . In this paper, the automorphism group of the graph C ⁢ ( G , X ) \mathcal{C}(G,X) is determined when G = PSL 2 ⁡ ( q ) G=\operatorname{PSL}_{2}(q) .

A Cayley digraph on a group G is called NNN if the Cayley digraph is normal and its automorphism group contains a non-normal regular subgroup isomorphic to G. A group is called an NNND-group or an NNN-group if there is an NNN Cayley digraph or graph on the group, respectively. In this paper, it is shown that there is no cyclic NNND-group, and hence no cyclic NNN-group. Furthermore, a dihedral group of order 2 n 2n is an NNND-group or an NNN-group if and only if n >= 6 n\geq 6 is even and n not equal 8 n eq 8 .

An Abelian group G is called strongly not divisible if nG not equal G for every integer n not equal +/- 1. In J. A. Lewallen and N. Sagullo, A note on OI torsion abelian groups, Missouri J. Math. Sci. 27 (2015), 1, 33-36, a characterization of the torsion strongly not divisible groups (referred to there as OI-groups) was obtained. In this note, we provide characterizations for broad classes of torsion-free and mixed strongly not divisible groups.

In a previous work, it was shown that there is a finitely generated metabeliangroup of finite torsion-free rank associated with each non-constant, non-unit integer poly-nomial. Here an algorithm is devised which is able to decide if two such groups are iso-morphic.

In this article, we study rational representations of groups of order p 5 p{5} , where p is an odd prime. For a p-group G and an irreducible complex character chi of G, the construction of an irreducible rational matrix representation of G affording the character Omega ( chi ) \Omega(\chi) is equivalent to determining a pair ( H , psi ) (H,\psi) , with H a subgroup of G and psi a linear character of H such that psi G = chi \psi{G}=\chi and Q ( psi ) = Q ( chi ) \mathbb{Q}(\psi)=\mathbb{Q}(\chi) , where Omega ( chi ) = m Q ( chi ) & sum; sigma is an element of Gal ( Q ( chi ) / Q ) chi sigma \Omega(\chi)=m_{\mathbb{Q}}(\chi)\sum_{\sigma\in\mathrm{Gal}(\mathbb{Q}(\chi)/\mathbb{Q})}\chi{\sigma} and m Q ( chi ) m_{\mathbb{Q}}(\chi) denotes the Schur index of chi over & Qopf;. For each inequivalent irreducible rational representation of every group of order p 5 p{5} , we determine such a pair, referred to as a required pair. We also derive combinatorial formulations for the Wedderburn decomposition of rational group algebras of these p-groups, using results from their rational representations.

Let G be a locally compact group. The Chabauty space S UB(G), consisting of the closed subgroups of G endowed with the Chabauty topology, provides a compact topological framework for understanding convergence phenomena among subgroups. In this paper, we investigate the Chabauty limits of two distinguished families. In the first part, we characterize locally compact groups that arise as Chabauty limits of their closed almost connected subgroups, showing that this occurs precisely when the quotient G/G(0) is compactly ruled. In the second part, we study the Chabauty closure of the set of open subgroups. For pro-Lie groups, we show that this closure coincides with the set of closed subgroups containing the identity component. We also provide a new characterization of totally disconnected locally compact groups in terms of the Chabauty limits of their open subgroups.

Let G be a permutation group, with minimal degree mu (G) and base size b(G). We show that there exists a universal constant c > 0 such that, for infinitely many n, there is a transitive permutation group G of degree n with mu (G) b(G) >= c center dot n(2). We also identify some classes of transitive and intransitive groups whose base size and minimal degree have a smaller upper bound, shared with primitive groups.

A natural generalization of Bring's curve, valid over any field of characteristic zero or characteristic p >= 7 p\geq 7 , is the algebraic variety V of PG ( m - 1 , K ) \operatorname{PG}(m-1,\mathbb{K}) , m >= 5 m\geq 5 , which is the complete intersection of the projective algebraic hypersurfaces of the homogeneous equations x 1 k + & ctdot; + x m k = 0 x_{1}{k}+\cdots+x_{m}{k}=0 with 1 <= k <= m - 2 1\leq k\leq m-2 . In positive characteristic, we also assume m <= p - 1 m\leq p-1 . Up to a change of coordinates in PG ( m - 1 , K ) \operatorname{PG}(m-1,\mathbb{K}) , we show that V is a projective, absolutely irreducible, non-singular curve of PG ( m - 2 , K ) \operatorname{PG}(m-2,\mathbb{K}) . We show that if the automorphism group Aut ( V ) \operatorname{Aut}(V) is tame (in particular in characteristic zero), then Aut ( V ) \operatorname{Aut}(V) is linear and isomorphic to Sym m \mathrm{Sym}_{m} . Remarkably, in positive characteristic, Aut ( V ) \operatorname{Aut}(V) may be larger; in particular, Aut ( V ) \operatorname{Aut}(V) happens to contain non-tame projectivities and this occurs precisely when m = p - 1 m=p-1 . In positive characteristic, we further investigate the two extremal cases, m = 5 m=5 and m = p - 1 m=p-1 , providing information on the number of rational points. In particular, for m = 5 m=5 , we show that there exist infinitely many primes p such that V is an F p 2 \mathbb{F}_{{p{2}}} -maximal curve of genus 4. We also highlight connections with previous work by R & eacute;dei on the famous Minkowski conjecture proven by Haj & oacute;s (1941), as well as with more recent results by Rodr & iacute;guez Villegas, Voloch, and Zagier (2001) on plane curves attaining the St & ouml;hr-Voloch bound, and the regular sequence problem for systems of diagonal equations introduced by Conca, Krattenthaler, and Watanabe (2009).

Let G be a locally compact group and let SUB(G) be the set of closed subgroups of G equipped with the Chabauty topology. In this paper, we establish some necessary and sufficient conditions for a map T : X -> SUB(G), from a topological space X into SUB(G), to be continuous.

We introduce a simple equivalence relation on strongly minimal sets in a structure of finite Morley rank, which corresponds, in stability theory, to the non-orthogonality of the associated types. We use it in a group G of finite Morley rank to define, for each strongly minimal set X, two connected normal subgroups M G ( X ) M_{G}(X) and W G ( X ) W_{G}(X) . When G is connected, these subgroups provide a central decomposition of G that yields a direct product decomposition of G / Z ( G ) G/Z(G) into unidimensional factors, as well as a central decomposition of its derived subgroup into unidimensional subgroups.

Let F n be a free group of rank n. An SL ( 2 ) character of F n means the trace of an SL ( 2 ) representation of F n. Let K be a field of characteristic 0. The automorphism group Aut ( F n ) naturally acts on the commutative K-algebra of the SL ( 2 ) characters. Then the augmentation ideal J n + of the commutative K-algebra generated by the SL ( 2 ) characters is an Aut ( F n )-invariant. The main purpose of this paper is to study the structure of the graded quotient ( J n + ) k / ( J n + ) k + 1 as an Aut ( F n )-module.

In an elementary way, we construct a family of pairwise non-embeddable torsion-free groups which contains 2 kappa 2<^>{\kappa} groups of cardinality kappa for every infinite cardinal number kappa up to the first strong limit cardinal of uncountable cofinality.

Recently, the author published a book [A. A. Ivanov, Ever-Evolving Groups-an Introduction to Modern Finite Group Theory, Algebr. Appl. 32, Springer, Cham, 2025] where he summarised the recent progress in the geometric theory of sporadic groups and outlined some geometries which require further investigation. Among them was a geometry for the smallest Conway sporadic simple group Co 3 \mathrm{Co}_{3} , with diagram originally introduced in [M. A. Ronan and G. Stroth, Minimal parabolic geometries for the sporadic groups, European J. Combin. 5 (1984), 1, 59-91] (cf. [M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs, London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316-331, Table 1] and [F. Buekenhout, Diagram geometries for sporadic groups, Finite Groups-Coming of Age (Montreal 1982), Contemp. Math. 45, American Mathematical Society, Providence (1985), geometry (23), p. 14]), which we denote by G ( Co 3 ) \mathcal{G}(\mathrm{Co}_{3}) . The 2-local geometries for the other Conway groups Co 1 \mathrm{Co}_{1} and Co 2 \mathrm{Co}_{2} are the tilde and Petersen geometries which have been intensively studied (cf. [A. A. Ivanov and S. V. Shpectorov, The flag-transitive tilde and Petersen-type geometries are all known, Bull. Amer. Math. Soc. (N. S.) 31 (1994), 2, 173-184]). However, G ( Co 3 ) \mathcal{G}(\mathrm{Co}_{3}) seems to be studied less. There is another geometry associated with Co 3 \mathrm{Co}_{3} (cf. [M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs, London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316-331, Table 1] and [F. Buekenhout, Diagram geometries for sporadic groups, Finite Groups-Coming of Age (Montreal 1982), Contemp. Math. 45, American Mathematical Society, Providence (1985), geometry (23), p. 14]) with diagram The elements of type 1, 2, 3, and 4 in the geometry G 276 \mathcal{G}_{276} correspond to cliques of size 1, 2, 3, and 6, respectively, in a double cover of the complete graph on 276 vertices. The group Co 3 \mathrm{Co}_{3} acts doubly transitively on the vertex set of the complete graph and flag-transitively on G 276 \mathcal{G}_{276} . This double cover is naturally associated with the well-known 2-graph of Co 3 \mathrm{Co}_{3} . The geometry G 276 \mathcal{G}_{276} is simply connected, as established by [M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs, London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316-331, Proposition 6]. In the present paper, we recover the distance 2 graph of G 276 \mathcal{G}_{276} from the universal cover of G ( Co 3 ) \mathcal{G}(\mathrm{Co}_{3}) , in particular re-proving the simple connectedness of the latter geometry originally established in [A. Chermak, B. Oliver and S. Shpectorov, The linking systems of the Solomon 2-local finite groups are simply connected, Proc. Lond. Math. Soc. (3) 97 (2008), 1, 209-238]. A key step in our proof makes use of the simple connectedness of the Petersen-type geometry of the McLaughlin group, as shown in [B. Baumeister, A. A. Ivanov and D. V. Pasechnik, A characterization of the Petersen-type geometry of the McLaughlin group, Math. Proc. Cambridge Philos. Soc. 128 (2000), 1, 21-44].

Two groups L 1 L_{1} and L 2 L_{2} are compatible if there exists a finite group G with isomorphic normal subgroups N 1 N_{1} and N 2 N_{2} such that L 1 congruent to G / N 1 L_{1}\cong G/N_{1} and L 2 congruent to G / N 2 L_{2}\cong G/N_{2} . In this paper, we give new necessary conditions for two groups to be compatible.

We study the orders of products of two class transpositions in the group CT ( Z ) \operatorname{CT}(Z) , a simple subgroup of the symmetric group on the integers. For pairs of class transpositions sharing a common vertex, we prove that the order of their product is either 1, 3, or infinity, and provide a precise criterion for the infinite order case. Furthermore, we investigate pairs of equal-residue and equal-modulus class transpositions, establishing conditions under which their product has finite or infinite order. Our results provide a partial answer to a question posed in the Kourovka notebook (see Question 18.48).