Abstract Let 𝐺 be a locally compact group and let S ⁢ U ⁢ B ⁢ ( G ) \mathcal{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 → S ⁢ U ⁢ B ⁢ ( G ) \mathbf{T}\colon X\to\mathcal{SUB}(G) , from a topological space 𝑋 into S ⁢ U ⁢ B ⁢ ( G ) \mathcal{SUB}(G) , to be continuous.

Abstract The weak commutativity group χ ⁢ ( G ) \chi(G) is generated by two isomorphic groups 𝐺 and G φ G^{\varphi} subject to the relations [ g , g φ ] = 1 [g,g^{\varphi}]=1 for all g ∈ G g\in G . We obtain new expressions for the terms of the derived series and the lower central series of χ ⁢ ( G ) \chi(G) . We also present new bounds for the exponent of some sections of χ ⁢ ( G ) \chi(G) .

Abstract Let F n F_{n} be a free group of rank 𝑛. An SL ⁢ ( 2 ) \mathrm{SL}(2) character of F n F_{n} means the trace of an SL ⁢ ( 2 ) \mathrm{SL}(2) representation of F n F_{n} . Let 𝐾 be a field of characteristic 0. The automorphism group Aut ⁢ ( F n ) \mathrm{Aut}(F_{n}) of F n F_{n} naturally acts on the commutative 𝐾-algebra of the SL ⁢ ( 2 ) \mathrm{SL}(2) characters. Then the augmentation ideal J n + J_{n}^{+} of the commutative 𝐾-algebra generated by the SL ⁢ ( 2 ) \mathrm{SL}(2) characters is an Aut ⁢ ( F n ) \mathrm{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 (J_{n}^{+})^{k}/(J_{n}^{+})^{k+1} as an Aut ⁢ ( F n ) \mathrm{Aut}(F_{n}) -module.

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

Abstract 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) .

Abstract Quandles are algebraic structures showing up in different mathematical contexts. A group 𝐺 with the conjugation operation forms a quandle, Conj ⁡ ( G ) \operatorname{Conj}(G) . In the opposite direction, a group As ⁡ ( Q ) \operatorname{As}(Q) can be constructed out of any quandle 𝑄. We explore As ⁡ ( Conj ⁡ ( G ) ) \operatorname{As}(\operatorname{Conj}(G)) for a group 𝐺 admitting a presentation with only conjugation and power relations. Symmetric groups S n S_{n} are typical examples. For such groups, we show that As ⁡ ( Conj ⁡ ( G ) ) \operatorname{As}(\operatorname{Conj}(G)) injects into G × Z m G\times\mathbb{Z}^{m} , where 𝑚 is the number of conjugacy classes of 𝐺. From this, we deduce information about the torsion, centre, and derived group of As ⁡ ( Conj ⁡ ( G ) ) \operatorname{As}(\operatorname{Conj}(G)) . As an application, we compute the second integral quandle homology group of Conj ⁡ ( S n ) \operatorname{Conj}(S_{n}) , and unveil rich torsion therein.

Abstract A string group generated by involutions, or SGGI, is a pair Γ = ( G , S ) \Gamma=(G,S) , where 𝐺 is a group and S = { ρ 0 , … , ρ r − 1 } S=\{\rho_{0},\ldots,\rho_{r-1}\} is an ordered set of involutions generating 𝐺 and satisfying the following commuting property: for all i , j ∈ { 0 , … , r − 1 } i,j\in\{0,\ldots,r-1\} , | i − j | ≠ 1 \lvert i-j\rvert\neq 1 implies ( ρ i ⁢ ρ j ) 2 = 1 (\rho_{i}\rho_{j})^{2}=1 . When 𝑆 is an independent set, the rank of Γ is the cardinality of 𝑆. We determine an upper bound for the rank of an SGGI over the alternating group of degree 𝑛. Our bound is tight when n ≡ 0 , 1 , 4 ⁢ ( mod ⁢ 5 ) n\equiv 0,1,4\ (\mathrm{mod}\ {5}) .

Abstract The prime graph , also called the Gruenberg–Kegel graph , of a finite group 𝐺 is the labelled graph Γ ⁢ ( G ) \Gamma(G) with vertices the prime divisors of | G | \lvert G\rvert and edges the pairs { p , q } \{p,q\} for which 𝐺 contains an element of order p ⁢ q pq . A group 𝐺 is recognisable by its prime graph if every group 𝐻 with Γ ⁢ ( H ) = Γ ⁢ ( G ) \Gamma(H)=\Gamma(G) is isomorphic to 𝐺. Cameron and Maslova have shown that every group that is recognisable by its prime graph is almost simple. This justifies the significant amount of attention that has been given to determining which simple or almost simple groups are recognisable by their prime graphs. This problem has been completely solved for certain families of simple groups, including the sporadic groups. A natural extension of the problem is to determine which groups are recognisable by their unlabelled prime graphs, i.e. by the isomorphism types of their prime graphs. Here we determine which of the sporadic finite simple groups are recognisable by the isomorphism types of their prime graphs. We also show that, for every sporadic group 𝐺 that is not recognisable by the isomorphism type of Γ ⁢ ( G ) \Gamma(G) , there are infinitely many groups 𝐻 with Γ ⁢ ( H ) ≅ Γ ⁢ ( G ) \Gamma(H)\cong\Gamma(G) .

Abstract Let 𝐹 be a finite field, and for any integer k ≥ 0 k\geq 0 , let p k p_{k} be the power function on 𝐹 defined by p k ⁢ ( x ) = x k p_{k}(x)=x^{k} . We determine the group of CCZ automorphisms of p k p_{k} , i.e. the group of invertible affine transformations which preserve the graph of p k p_{k} .

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 O 2 ⁢ ( G ) > 1 \mathbf{O}_{2}(G)>1 and the case O 2 ⁢ ( G ) = 1 \mathbf{O}_{2}(G)=1 . Specifically we prove that if O 2 ⁢ ( G ) > 1 \mathbf{O}_{2}(G)>1 , then 𝐺 is a { 2 , p } \{2,p\} -group. Finally, by taking into consideration the examples presented in the analysis of the O 2 ⁢ ( G ) = 1 \mathbf{O}_{2}(G)=1 case, we deduce some interesting and unexpected results about the connectedness of the real prime graph Γ R ⁢ ( G ) \Gamma_{\mathbb{R}}(G) . In particular, we find that there are groups such that Γ R ⁢ ( G ) \Gamma_{\mathbb{R}}(G) has 3 or 4 connected components.