In this paper we introduce and study a family $\mathcal{A}_n(q)$ of abelian subgroups of $\GL_n(q)$ covering every element of $\GL_n(q)$. We show that $\mathcal{A}_n(q)$ contains all the centralisers of cyclic matrices and equality holds if $q>n$. Also, for $q>2$, we prove a simple closed formula for the size of $\mathcal{A}_n(q)$ and give an upper bound if $q=2$. A subset $X$ of a finite group $G$ is said to be pairwise non-commuting if $xy\not=yx$, for distinct elements $x, y$ in $X$. As an application of our results on $\mathcal{A}_n(q)$, we prove lower and upper bounds for the maximum size of a pairwise non-commuting subset of $\GL_n(q)$. (This is the clique number of the non-commuting graph.) Moreover, in the case where $q>n$, we give an explicit formula for the maximum size of a pairwise non-commuting set.
The commuting graph of a non-abelian group is a simple graph in which the vertices are the non-central elements of the group, and two distinct vertices are adjacent if and only if they commute. In this paper, we classify (up to isomorphism) all finite non-abelian groups whose commuting graphs are acyclic, planar or toroidal. We also derive explicit formulas for the genus of the commuting graphs of some well-known class of finite non-abelian groups, and show that, every collection of finite non-abelian groups whose commuting graphs have the same genus is finite.
Let G be a group. A subset X of G is a set of pairwise non-commuting elements if xy is not equal to yx for any two distinct elements x and y in X. If |X|>=|Y| for any other set of pairwise non-commuting elements Y in G, then X is said to be a maximal subset of pairwise non-commuting elements. In this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements for finite minimal non-abelian groups.
In this paper, we initiate the study of spectrum of the commuting graphs of finite non-abelian groups. We first compute the spectrum of this graph for several classes of finite groups, in particular AC-groups. We show that the commuting graphs of finite non-abelian AC-groups are integral. We also show that the commuting graph of a finite non-abelian group $G$ is integral if $G$ is not isomorphic to the symmetric group of degree $4$ and the commuting graph of $G$ is planar. Further it is shown that the commuting graph of $G$ is integral if the commuting graph of $G$ is toroidal.
Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph Γ(G) of G is a graph with vertex set is non central elements of G and two vertices x, y are adjacent if and only if they are commute. In this paper we calculate the energy, Laplacian energy and spectrum of non-commuting graph of dihedral group D2n. Also we will obtain the energy of non-commuting graph of D2n \times D2n and G \times H, where G is a non-abelian finite group and H is an abelian finite group
Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. The non-commuting graph $Γ_G$ associated to $G$ is the graph whose vertex set is $G\setminus Z(G)$ and two distinct elements $x,y$ are adjacent if and only if $xy\neq yx$. We prove that if $G$ and $H$ are non-abelian nilpotent groups with irregular isomorphic non-commuting graphs, then $|G|=|H|$.
Let $G$ be a nonabelian group. We say that $G$ has an abelian partition, if there exists a partition of $G$ into commuting subsets $A_1, A_2, \ldots, A_n$ of $G$, such that $|A_i|\geqslant 2$ for each $i=1, 2, \ldots, n$. This paper investigates problems relating to group with abelian partitions. Among other results, we show that every finite group is isomorphic to a subgroup of a group with an abelian partition and also isomorphic to a subgroup of a group with no abelian partition. We also find bounds for the minimum number of partitions for several families of groups which admit abelian part…
Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that intersects every conjugacy class of involutions of G.
A finite non-abelian group $G$ is called commuting integral if the commuting graph of $G$ is integral. In this paper, we show that a finite group is commuting integral if its central factor is isomorphic to ${\mathbb{Z}}_p \times {\mathbb{Z}}_p$ or $D_{2m}$, where $p$ is any prime integer and $D_{2m}$ is the dihedral group of order $2m$.
In this paper, we compute the Laplacian spectrum of non-commuting graphs of some classes of finite non-abelian groups. Our computations reveal that the non-commuting graphs of all the groups considered in this paper are L-integral. We also obtain some conditions on a group $G$ so that its non-commuting graph is L-integral.
For a non-abelian group $G$, its commuting conjugacy class graph $\mathcal{CCC}(G)$ is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x' \in x^G$ and $y' \in y^G$ such that $x'y' = y'x'$. In this paper we compute the genus of $\mathcal{CCC}(G)$ for six well-known classes of non-abelian two-generated groups (viz. $D_{2n}, SD_{8n}, Q_{4m}, V_{8n}, U_{(n, m)}$ and $G(p, m, n)$) and determine whether $\mathcal{CCC}(G)$ for these groups are planar, tor…
Let $G$ be a nonabelian group, $A\subseteq G$ an abelian subgroup and $n\geqslant 2$ an integer. We say that $G$ has an $n$-abelian partition with respect to $A$, if there exists a partition of $G$ into $A$ and $n$ disjoint commuting subsets $A_1, A_2, \ldots, A_n$ of $G$, such that $|A_i|>1$ for each $i=1, 2, \ldots, n$. We first classify all nonabelian groups, up to isomorphism, which have an $n$-abelian partition for $n=2, 3$. Then, we provide some formulas concerning the number of spanning trees of commuting graphs associated with certain finite groups. Finally, we point out some ways to f…
Let $G$ be a finite non-abelian group and $Γ_{nc}(G)$ be its non-commuting graph. In this paper, we compute spectrum and energy of $Γ_{nc}(G)$ for certain classes of finite groups. As a consequence of our results we construct infinite families of integral complete $r$-partite graphs. We compare energy and Laplacian energy (denoted by $E(Γ_{nc}(G))$ and $LE(Γ_{nc}(G))$ respectively) of $Γ_{nc}(G)$ and conclude that $E(Γ_{nc}(G)) \leq LE(Γ_{nc}(G))$ for those groups except for some non-abelian groups of order $pq$. This shows that the conjecture posed in [Gutman, I., Abreu, N. M. M., Vinagre, C.…
Given a finite group $G$ and a subset $X$ of $G$, the commuting graph of $G$ on $X$, denoted by ${\cal C}(G,X)$, is the graph that has $X$ as its vertex set with $x,y\in X$ joined by an edge whenever $x\neq y$ and $xy=yx$. Let $T$ be a transversal of the center $Z(G)$ of $G$. When $G$ is a finite non-abelian group and $X=T\setminus Z(G)$, we denote the graph ${\cal C}(G,X)$ by ${\cal T}(G)$. In this paper, we show that ${\cal T}(G)$ is a connected strongly regular graph if and only if $G$ is isoclinic to an extraspecial $2$-group of order at least $32$. We also characterize the finite non-abel…
In this paper we compute common neighbourhood (abbreviated as CN) spectrum and energy of commuting conjugacy class graph of several families of finite non-abelian groups. As a consequence of our results we show that the commuting conjugacy class graphs of the groups $D_{2n}$, $T_{4n}$, $SD_{8n}$, $U_{(n,m)}$, $U_{6n}$, $V_{8n}$, $G(p, m, n)$ and some families of groups whose central quotient is isomorphic to $D_{2n}$ or $Z_p \times Z_p$, for some prime $p$, are CN-integral but not CN-hyperenergetic.
In 2004, Csörgő constructed a loop of nilpotency class three with abelian group of inner mappings. Until now, no other examples were known. We construct many such loops from groups of nilpotency class two by replacing the product $xy$ with $xyh$ in certain positions, where $h$ is a central involution. The location of the replacements is ultimately governed by a symmetric trilinear alternating form.
Let $G$ be a group and $Z(G)$ be its center. We associate a commuting graph $Γ(G)$, whose vertex set is $G\setminus Z(G)$ and two distinct vertices are adjacent if they commute. We say that $Γ(G)$ is strong $k$ star free if the $k$ star graph is not a subgraph of $Γ(G)$. In this paper, we characterize all strong $5$ star free commuting graphs. As a byproduct, we classify all strong claw-free graphs. Also, we prove that the set of all non-abelian groups whose commuting graph is strong $k$ star free is finite.
A finite non-abelian group $G$ is called super integral if the spectrum, Laplacian spectrum and signless Laplacian spectrum of its commuting graph contain only integers. In this paper, we first compute various spectra of several families of finite non-abelian groups and conclude that those groups are super integral. As an application of our results we obtain some positive integers $n$ such that $n$-centralizer groups are super integral. We also obtain some positive rational numbers $r$ such that $G$ is super integral if it has commutativity degree $r$. In the last section, we show that $G$ is …
We study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Gamma is any graph not containing a four-cycle without chords, then the group G(Gamma) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Gamma) does not have a subgroup isomorphic to a direct product of non-abelian free groups. We also obtain corresponding and more general results for monoids.
The set of all centralizers of elements in a finite group $G$ is denoted by $Cent(G)$ and $G$ is called $n-$centralizer if $|Cent(G)| = n$. In this paper, the structure of centralizers in a non-abelian finite group $G$ with this property that $\frac{G}{Z(G)} \cong Z_{p^2} \rtimes Z_{p^2}$ is obtained. As a consequence, it is proved that such a group has exactly $[(p+1)^2+1]$ element centralizers and the structure of the commuting conjugacy class graph of $G$ is completely determined.
In this work we study the structure and cardinality of maximal sets of commuting and anticommuting Paulis in the setting of the abelian Pauli group. We provide necessary and sufficient conditions for anticommuting sets to be maximal, and present an efficient algorithm for generating anticommuting sets of maximum size. As a theoretical tool, we introduce commutativity maps, and study properties of maps associated with elements in the cosets with respect to anticommuting minimal generating sets. We also derive expressions for the number of distinct sets of commuting and anticommuting abelian Pau…
Let $G$ be a non-abelian group. The non-commuting graph $\mathcal{A}_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joint if and only if they do not commute. In a finite simple graph $Γ$ the maximum size of a complete subgraph of $Γ$ is called the clique number of $Γ$ and it is denoted by $ω(Γ)$. In this paper we characterize all non-solvable groups $G$ with $ω(\mathcal{A}_G)\leq 57$, where the number 57 is the clique number of the non-commuting graph of the projective special linear group $\mathrm{PSL}(2,7)$. We also complete the det…
We study rack and quandle coverings from a universal algebraic viewpoint and we show how they can be understood using the notion of strongly abelian congruences. We provide an abstract characterization of several particular types of covering extensions, such as central and abelian ones. We give a new characterization of simply connected quandles and we show that the categorical notion of normal extension coincides with the notion of central covering. We answer several questions from the papers of Clark, Saito and Vendramin \cite{CS} and \cite{CSV} about identities preserved by quandle covering…
Let αbe an automorphism of the totally disconnected group G. The compact open subgroup, V, if G is tidy for αif [α(V') : α(V')\cap V'] is minimised at V, where V' ranges over all compact open subgroups of G. Identifying a subgroup tidy for αis analogous to identifying a basis which puts a linear transformation into Jordan canonical form. This analogy is developed here by showing that commuting automorphisms have a common tidy subgroup of G and, conversely, that a group H of automorphisms having a common tidy subgroup V is abelian modulo the automorphisms which leave V invariant. Certain subgro…
A hypermap is an embedding of a connected hypergraph into an orientable closed surface. A covering between hypermaps is a homomorphism between the embedded hypergraphs which extends to an orientation-preserving covering of the supporting surfaces. A covering of a hypermap onto itself is an automorphism, and a hypermap is regular if its automorphism group acts transitively on the brins. Depending on the algebraic theory of regular hypermaps and hypermap operations, the abelian regular coverings over the quaternion hypermap are investigated. We define normalized multicyclic coverings between reg…
We classify the orientably regular maps which are elementary abelian regular branched coverings of Platonic maps M, in the case where the covering group and the rotation group G of M have coprime orders. The method involves studying the representations of G on certain homology groups of the sphere, punctured at the branch-points. We give a complete classification for branching over faces (or, dually, vertices) of M, and outline how the method extends to other branching patterns.
The commuting graph $Δ(G)$ of a finite non-abelian group $G$ is a simple graph with vertex set $G$ and two distinct vertices $x, y$ are adjacent if $xy = yx$. In this paper, among some properties of $Δ(G)$, we investigate $Δ(SD_{8n})$ the commuting graph of the semidihedral group $SD_{8n}$. In this connection, we discuss various graph invariants of $Δ(SD_{8n})$ including minimum degree, vertex connectivity, independence number, matching number and detour properties. We also obtain the Laplacian spectrum, metric dimension and resolving polynomial of $Δ(SD_{8n})$.
We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups, where the hypothesis of residual finiteness is always satisfied). We also show that, for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero.
Let $G=A_n$, a finite alternating group. We study the commuting graph of $G$ and establish, for all possible values of $n$ barring $13, 14, 17$ and $19$, whether or not the independence number is equal to the clique-covering number.
We study the question how many subgroups, cosets or subspaces are needed to cover a finite Abelian group or a vector space if we have some natural restrictions on the structure of the covering system. For example we determine, how many cosets we need, if we want to cover all but one element of an Abelian group. This result is a group theoretical extension of the theorem of Brouwer, Jamison and Schrijver about the blocking number of an affine space. We show that these covering problems are closely related to combinatorial problems, including the so called additive basis conjecture, the three-fl…
The commuting conjugacy class graph of a non-abelian group $G$, denoted by $\mathcal{CCC}(G)$, is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x' \in x^G$ and $y' \in y^G$ such that $x'y' = y'x'$. In this paper we compute various spectra and energies of commuting conjugacy class graph of the groups $D_{2n}, Q_{4m}, U_{(n, m)}, V_{8n}$ and $SD_{8n}$. Our computation shows that $\mathcal{CCC}(G)$ is super integral for these groups. We compare vari…
In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if $G$ and $H$ are finite groups with isomorphic non-commuting graphs and $G$ is nilpotent, then $H$ must be nilpotent as well (Conjecture 2). We pose a new conjecture (Conjecture 3) that, together with the assumption $|Z(G)|\geq|Z(H)|$, implies Conjecture 2 and we prove it for groups in which all centralizers of non-central elements are abelian.
We give a simple argument to show that if α is an affine isometric action of a product G x H of topological groups on a reflexive Banach space X with linear part π, then either π(H) fixes a unit vector or α|G almost fixes a point on X. It follows that any affine isometric action of an abelian group on a reflexive Banach space X, whose linear part fixes no unit vectors, almost fixes points on X.
We show that, if $w_1, \ldots , w_6$ are words which are not an identity of any (non-abelian) finite simple group, then $w_1(G)w_2(G) \cdots w_6(G) = G$ for all (non-abelian) finite simple groups $G$. In particular, for every word $w$, either $w(G)^6 = G$ for all finite simple groups, or $w(G)=1$ for some finite simple groups. These theorems follow from more general results we obtain on characteristic collections of finite groups and their covering numbers, which are of independent interest and have additional applications.
Let $G$ be an algebraic group. For $d\geq 1$, we define the commuting probabilities $cp_d(G) = \frac{dim(\mathfrak C_d(G))}{dim(G^d)}$, where $\mathfrak C_d(G)$ is the variety of commuting $d$-tuples in $G$. We prove that for a reductive group $G$ when $d$ is large, $cp_d(G)\sim \fracα{n}$ where $n=\dim(G)$, and $α$ is the maximal dimension of an Abelian subgroup of $G$. For a finite reductive group $G$ defined over the field $\mathbb F_q$, we show that $cp_{d+1}(G(\mathbb F_q))\sim q^{(α-n)d}$, and give several examples.
The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P \subset (0,1] be the set of commuting probabilities of all finite groups. We prove that every point of P is nearly an Egyptian fraction of bounded complexity. As a corollary we deduce two conjectures of Keith Joseph from 1977: all limit points of P are rational, and P is well ordered by >. We also prove analogous theorems for bilinear maps of abelian groups.
We describe a large-scale project in applied automated deduction concerned with the following problem of considerable interest in loop theory: If $Q$ is a loop with commuting inner mappings, does it follow that $Q$ modulo its center is a group and $Q$ modulo its nucleus is an abelian group? This problem has been answered affirmatively in several varieties of loops. The solution usually involves sophisticated techniques of automated deduction, and the resulting derivations are very long, often with no higher-level human proofs available.
Given a free product $G$, we investigate the existence of faithful free representations of the outer automorphism group $\text{Out}(G)$, or in other words of embeddings of $\text{Out}(G)$ into $\text{Out}\left(F_m\right)$ for some $m$. This is based on a work of Bridson and Vogtmann in which they construct embeddings of $\text{Out}\left(F_n\right)$ into $\text{Out}\left(F_m\right)$ for some values of $n$ and $m$ by interpreting $\text{Out}\left(F_n\right)$ as the group of homotopy equivalences of a graph $X$ of genus $n$, and by lifting homotopy equivalences of $X$ to a characteristic abelian …
We present a family of finite, non-abelian groups and propose that there are members of this family whose commuting graphs are connected and of arbitrarily large diameter. If true, this would disprove a conjecture of Iranmanesh and Jafarzadeh. While unable to prove our claim, we present a heuristic argument in favour of it. We also present the results of simulations which yielded explicit examples of groups whose commuting graphs have all possible diameters up to and including 10. Previously, no finite group whose commuting graph had diameter greater than 6 was known.
Let $G$ be a non-abelian group and $Z(G)$ be its center. The non-commuting graph $\mathcal{A}_G$ of $G$ is the graph whose vertex set is $G\backslash Z(G)$ and two vertices are joined by an edge if they do not commute. Let $\mathrm{SL}(2,q)$ be the special linear group of degree 2 over the finite field of order $q$. In this paper we prove that if $G$ is a group such that $\mathcal{A}_G\cong \mathcal{A}_{\mathrm{SL}(2,q)}$ for some prime power $q\geq 2$, then $G\cong \mathrm{SL}(2,q)$.
For a finite group $G$, let $d(G)$ denote the probability that a randomly chosen pair of elements of $G$ commute. We prove that if $d(G)>1/s$ for some integer $s>1$ and $G$ splits over an abelian normal nontrivial subgroup $N$, then $G$ has a nontrivial conjugacy class inside $N$ of size at most $s-1$. We also extend two results of Barry, MacHale, and N\'ı Shé on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if $d(G)>5/16$ then either $G$ is supersolvable, or $G$ isoclinic to $A_4$, or $G/\Center(G)$ is isoclinic to $A_4$.
We classify the nonsplit extensions of elementary abelian $p$-groups by $PSL_2(q)$, with odd $p$ dividing $q-1$, for an irreducible induced action, calculate the relevant low-dimensional cohomology groups, and describe the automorphism groups of such extensions.
We prove that any product of two non-abelian free groups, $Γ=\mathbb F_m\times\mathbb F_k$, for $m,k\geq 2$, is not Hilbert-Schmidt stable. This means that there exist asymptotic representations $π_n:Γ\rightarrow \text{U}({d_n})$ with respect to the normalized Hilbert-Schmidt norm which are not close to actual representations. As a consequence, we prove the existence of contraction matrices $A,B$ such that $A$ almost commutes with $B$ and $B^*$, with respect to the normalized Hilbert-Schmidt norm, but $A,B$ are not close to any matrices $A',B'$ such that $A'$ commutes with $B'$ and $B'^*$. Thi…
Given a finite group $G$, we say that $G$ has weak normal covering number $γ_w(G)$ if $γ_w(G)$ is the smallest integer with $G$ admitting proper subgroups $H_1,\ldots,H_{γ_w(G)}$ such that each element of $G$ has a conjugate in $H_i$, for some $i\in \{1,\ldots,γ_w(G)\}$, via an element in the automorphism group of $G$. We prove that the weak normal covering number of every non-abelian simple group is at least $2$ and we classify the non-abelian simple groups attaining $2$. As an application, we classify the non-abelian simple groups having normal covering number $2$. We also show that the weak…
Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. Associate a graph $Γ_G$ (called non-commuting graph of $G$) with $G$ as follows: take $G\setminus Z(G)$ as the vertices of $Γ_G$ and join two distinct vertices $x$ and $y$, whenever $xy\neq yx$. Here, we prove that "the commutativity pattern of a finite non-abelian $p$-group determine its order among the class of groups"; this means that if $P$ is a finite non-abelian $p$-group such that $Γ_P\cong Γ_H$ for some group $H$, then $|P|=|H|$.
For any (Hausdorff) compact group $G$ with the normalized Haar measure ${\mathbf m}_G$, denote by ${\rm cp}(G)$ the probability ${\mathbf m}_{G\times G}(\{(x,y)\in G\times G \;|\; xy=yx\})$ of commuting a randomly chosen pair of elements of $G$. Here we prove that if ${\rm cp}(G)>0$, then there exists a finite group $H$ such that ${\rm cp}(G)= \frac{{\rm cp}(H)}{|G:F|^2}$, where $F$ is the FC-center of $G$ i.e. the set of all elements of $G$ whose conjugacy classes are finite and $H$ is isoclinic to $F$ with ${\rm cp}(F)={\rm cp}(H)$. The latter equality enables one to transfer many existing r…
We develop a general theory of "bisets": sets with two commuting group actions. They naturally encode topological correspondences. Just as van Kampen's theorem decomposes into a graph of groups the fundamental group of a space given with a cover, we prove analogously that the biset of a correspondence decomposes into a "graph of bisets": a graph with bisets at its vertices, given with some natural maps. The "fundamental biset" of the graph of bisets recovers the original biset. We apply these results to decompose the biset of a Thurston map (a branched self-covering of the sphere whose critica…
We consider orbifoldings of the Moonshine Module with respect to the abelian group generated by a pair of commuting Monster group elements with one of prime order $p=2,3,5,7$ and the other of order $pk$ for $k=1$ or $k$ prime. We show that constraints arising from meromorphic orbifold conformal field theory allow us to demonstrate that each orbifold partition function with rational coefficients is either constant or is a hauptmodul for an explicitly found modular fixing group of genus zero. We thus confirm in the cases considered the Generalised Moonshine conjectures for all rational modular f…
