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Let G be a non-abelian group. The non-commuting graph [Formula: see text] 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 complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with [Formula: see text], where 57 is the clique number of the non-commuting graph of the projective special linear group PSL (2,7). We also determine [Formula: see text] for all finite minimal simple groups G.

We associate a graph [Formula: see text] to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\ Cyc (G) as vertex set, where Cyc (G) = {x ∈ G | 〈x,y〉 is cyclic for all y ∈ G} is called the cyclicizer of G, and join two vertices if they do not generate a cyclic subgroup. For a simple graph Γ, w(Γ) denotes the clique number of Γ, which is the maximum size (if it exists) of a complete subgraph of Γ. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group G is solvable whenever [Formula: see text] and the equality for a non-solvable group G holds if and only if G/ Cyc (G) ≅ A5 or S5.

In this article we prove that a set of points B of PG(n, 2) is a minimal blocking set if and only if ⟨B⟩ = PG(d, 2) with d odd and B is a set of d + 2 points of PG(d, 2) no d + 1 of them in the same hyperplane. As a corollary to the latter result we show that if G is a finite 2-group and n is a positive integer, then G admits a ℭ n+1-cover if and only if n is even and G≅ (C 2) n , where by a ℭ m -cover for a group H we mean a set 𝒞 of size m of maximal subgroups of H whose set-theoretic union is the whole H and no proper subset of 𝒞 has the latter property and the intersection of the maximal subgroups is core-free. Also for all n 0 an integer and p a prime number) for which there is a blocking set B of size n in PG(m,p) such that ⟨B⟩ = PG(m,p).

We associate a graph Γ G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\ Cyc(G) as vertex set, where Cyc(G) = {x ∊ G| 〈x, y〉 is cyclic for all y ∊ G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ G is finite if and only if Γ G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ G ≅ Γ H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are “unique”, i.e., if Γ G ≅ Γ H for some group H, then G ≅ H. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if Γ G ≅ Γ H for some group H, then |G| = |H|. These suggest the...

A cover for a group G is a collection of proper subgroups whose union is the whole group G. A cover is irredundant if no proper sub-collection is also a cover, and is called maximal if all its members are maximal subgroups. For an integer n > 2, a cover with n members is called an n-cover. Also, we denote σ (G) = n if G has an n-cover and does not have any m-cover for each integer m

Let n be an integer greater than 1. A group G is said to be n-rewritable (or a Qn-group) if for every n elements x1, x2,…,xn in G there exist distinct permutations σ and τ in Sn such that xσ(1)xσ (2) ⋅⋅⋅ xσ(n) = xτ(1)xτ(2) ⋅⋅⋅ xτ(n). In this paper, we characterize all 3-rewritable nilpotent 2-groups of class 2. Also we have found a bound for the nilpotency class of certain nilpotent 3-rewritable groups, and have shown that 3-rewritable groups satisfy a certain law.

We use Ramsey's theorem to generalise a result of L. Babai and T.S. Sós on Sidon subsets and then use this to prove that for an integer n > 1 the class of groups in which every infinite subset contains a rewritable n-subset coincides with the class of groups in which ever infinite subset contains n mutually disjoint non-empty subsets X1,…,Xn such that X1…Xn∩Xσ(1) …xσ(n) ≠ θ for some non-identity permutation σ on the set {1,…,n}.

Let n > 1 be an integer. A group G is said to be n-rewritable, whenever for any subset {x 1, …, x n } of elements of G, there exist distinct permutations τ, σ of the set {1, 2, …, n} such that x τ(1) … x τ(n) = x σ (1) … x σ (n). In this paper we show that an infinite group G is n-rewritable if and only if for every n infinite subsets X 1, …, X n of G there exist distinct permutations τ, σ of the set {1, 2, …, n} such that X τ (1) … X τ(n)∩ X σ(1) … X σ(n) ≠ ∅.

Let n be an integer greater than 1. A group G is said to be n -permutable whenever for every n -tuple ( x 1 ,…, x n ) of elements of G there exists a non-identity permutation σ of {1,…, n } such that x 1 ··· x n = x σ(1) ··· x σ( n ) . In this paper we prove that an infinite group G is n -permutable if and only if for every n infinite subsets X 1 ,…, X n of G there exists a non-identity permutation σ on {1,…, n } such that X 1 ··· X n ∪ X σ(1) ··· X σ( n ) ≠ ∅ .

In this paper we prove the following:1.Let m ≥ 2, n ≥ 1 be integers and let G be a group such that (XY)n = (YX)n for all subsets X, Y of size m in G. Thena)G is abelian or a BFC-group of finite exponent bounded by a function of m and n.b)If m ≥ n then G is abelian or |G| is bounded by a function of m and n.2.The only non-abelian group G such that (XY)2 = (YX)2 for all subsets X, Y of size 2 in G is the quaternion group of order 8.3.Let m, n be positive integers and G a group such that for all subsets Xi of size m in G. Then G is n-permutable or |G| is bounded by a function of m and n.

A complete classification is given of all groups G with the property that every automorphism centralizes a subgroup with prime index in G.

A group in which every element commutes with its endomorphic images is called an $E$-group. Our main result is that all 3-generator $E$-groups are abelian. It follows that the minimal number of generators of a finitely generated non-abelian $E$-group is four.

A group in which every element commutes with its endomorphic images is called an $E$-group. If $p$ is a prime number, a $p$-group $G$ which is an $E$-group is called a $pE$-group. Every abelian group is obviously an $E$-group. We prove that every 2-generator $E$-group is abelian and that all 3-generator $E$-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator $E$-group is abelian. We conjecture that every finite 3-generator $E$-group should be abelian. Moreover we show that the minimum order of a non-abelian $pE$-group is $p^8$ for any odd prime number $p$ and this order is $2^7$ for $p=2$. Some of these results are proved for a class wider than the class of $E$-groups.

Ore (1942) studied the automorphisms of finite monomial groups and Holmes (1956, pp. 23–93) has given the form of the automorphisms of the restricted monomial groups in the infinite case. The automorphism group of a standard wreath product has been studied by Houghton (1962) and Segal (1973, Chapter 4). Monomial groups and standard wreath products are both special cases of permutational wreath product. Here we investigate the automorphisms of the permutational wreath product and consider to what extent the results holding in the special cases remain true for the general construction. Our results extend those of Bunt (1968).

To study the significance of the recent convictions in the application of multimodal tools in teaching vocabularies, the present study sets out to investigate the extent of vocabulary learning within two different environments: audio-visual signs accompanied by intersemiotic translation (subtitles) versus audio-visual signs (no subtitles). Elementary level Participants watched the two different versions of The Lorax, a famous cartoon in Iran. To evaluate the effect of each set of signs on their vocabulary learning, participants were required to answer three levels of questions. The obtained results were analysed using paired sample t-test. The results revealed a better performance of the group who received subtitles.

This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: “Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?” We find an exponential function f(m,n) such that every such group G is Abelian whenever |G| > f(m,n) and this function can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble.