Journal of Group Theory

Abstract Let 𝐺 be a permutation group, with minimal degree <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>μ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mu(G) and base size <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo…

Abstract Let 𝐺 be a finite group and 𝑝 a prime. We denote by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">C</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathscr{C}_{p}(G) the poset of all cosets of 𝑝-subgroups of 𝐺. We c…

Abstract Let 𝐺 be a locally compact group and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">S</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">U</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">B</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathcal{SUB}(…

Abstract The weak commutativity group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>χ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \chi(G) is generated by two isomorphic groups 𝐺 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>G</m:mi> <m:mi>φ</m:mi> </m:…

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> F_{n} be a free group of rank 𝑛. An <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>SL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{SL}(2) character…

Abstract Given a finite group 𝐺 and a conjugacy class of involutions 𝑋 of 𝐺, we define the commuting involution graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">C</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathcal{C}(G,X)…

Abstract In an elementary way, we construct a family of pairwise non-embeddable torsion-free groups which contains <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mn>2</m:mn> <m:mi>κ</m:mi> </m:msup> </m:math> 2^{\kappa} groups of cardinality 𝜅 for every infinite cardinal number 𝜅 up to the first strong limit cardinal of uncountable cofinality.

Abstract A string group generated by involutions, or SGGI, is a pair <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>S</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \Gamma=(G,S) , where 𝐺 is a group and <m:math xmlns:m="http://www.w…

Abstract Quandles are algebraic structures showing up in different mathematical contexts. A group 𝐺 with the conjugation operation forms a quandle, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Conj</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{Conj}(G) . In the oppo…

Abstract We study the finite solvable groups 𝐺 in which every real element has prime power order. We divide our examination into two parts: the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="bold">O</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mr…

Abstract Let 𝐹 be a finite field, and for any integer <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> k\geq 0 , let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>p</m:mi> <m:mi>k</m:mi> </m:msub> </m:math> p_{k} be the power function on 𝐹 defined by <m:math xmlns:m="http://www.w3.org/1998/Math…

Abstract The prime graph , also called the Gruenberg–Kegel graph , of a finite group 𝐺 is the labelled graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \Gamma(G) with vertices the prime divisors of <m:math xml…

Abstract 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:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> …

Abstract By strengthening known results about primitivity-blocking words in free groups, we prove that, for any element 𝑤 of a free group of finite rank, there are words that cannot be subwords of any cyclically reduced automorphic image of 𝑤. This has implications for the average-case complexity of a variant of Whitehead’s problem.

Abstract 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 <m:math…

Abstract A cyclic complementary extension of a finite group 𝐴 is a finite group 𝐺 which contains 𝐴 and a cyclic subgroup 𝐶 such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>A</m:mi> <m:mo>∩</m:mo> <m:mi>C</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mn>1</m:mn> <m:mi>G</m:mi> </m:msub> <m:mo stretchy="false">}</m:mo…

Abstract We study the orders of products of two class transpositions in the group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>CT</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>Z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{CT}(Z) , a simple subgroup of the symmetric group on the integers. For pairs of class trans…

Abstract Two groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> L_{1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> L_{2} are compatible if there exists a finite group 𝐺 with isomorphic normal subgroups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <…

Abstract We introduce the special and general projectivity groups attached to a simplex 𝐹 of a thick, irreducible, spherical building of simply laced type. If the residue of 𝐹 is irreducible, we determine the permutation group of both projectivity groups of 𝐹, acting on the residue of 𝐹 and show that the special projectivity group determines the precise action of the Levi subgroup of a parabo…

Abstract We completely classify fixed point subgroups in torus knot groups, that is, groups of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">⟨</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo fence="t…