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Let Γ be a simple finite graph with vertex set V (Γ) and edge set E (Γ) . Let R be an equivalence relation on V (Γ) . The R -super Γ graph Γ R is a simple graph with vertex set V (Γ) and two distinct vertices are adjacent if either they are in the same R -equivalence class or there are elements in their respective R -equivalence classes that are adjacent in the original graph Γ . We first show that Γ R is a generalized join of some complete graphs and using this we obtain the adjacency and Laplacian spectrum of conjugacy super commuting graphs and order super commuting graphs of dihedral group D 2 n (n ≥ 3) , generalized quaternion group Q 4 m (m ≥ 2) and the nonabelian group Z p ⋊ Z q of order pq, where p and q are distinct primes with q | (p - 1) . [ABSTRACT FROM AUTHOR]

Given a finite group G, denote by cs(G) the set of the sizes of the conjugacy classes of G and by Cent(G) the set of the centralizers of elements of G. Consider a prime p and integers s ≥ 2 and n ≥ 2 , with gcd (p , s) = 1 . In this paper, some relations between cs(G) and | Cent (G) | are established in the case where cs (G) = { 1 , p n , p n − 1 s } . Further, when p ∈ { 2 , 3 } , we determine the values of s and the structure of a finite group G such that cs (G) = { 1 , p n , p n − 1 s } . We also describe the structure of an AC-group G such that [ G : Z (G) ] = 3 n s and | Cent (G) | = 1 + ∑ i = 0 n 3 i , where gcd (3 , s) = 1 . [ABSTRACT FROM AUTHOR]

Suppose G is a finite non-abelian group and (G) is a graph with non-central conjugacy classes of G as its vertex set. Two vertices L and K in such that ab = ba. This graph is called the commuting conjugacy class graph of G. The purpose of this paper is to compute the commuting conjugacy class graph of the finite 2 such that ab = ba. This graph is called the commuting conjugacy class graph of G. The purpose of this paper is to compute the commuting conjugacy class graph of the finite 2-group Gm(n) and G(n). [ABSTRACT FROM AUTHOR]

Let m be a cube-free positive integer and let p be a prime such that p ∤ m. In this paper, we find the number of conjugacy classes of completely reducible solvable cube-free subgroups in GL(2,q) of order m, where q is a power of p. [ABSTRACT FROM AUTHOR]

One of the important problem in finite groups theory is group characterization by specific property. Properties, such as element order, the set of element with the same order, etc. In this paper, we prove that Ree group ²G2(q), where q±√3q+1 is a prime number can be uniquely determined by its order and one conjugacy class size. [ABSTRACT FROM AUTHOR]

Let p be a prime and let G be a finite group such that the smallest prime that divides |G| is p. We find sharp bounds, depending on p, for the commuting probability and the average character degree to guarantee that G is nilpotent or supersolvable. [ABSTRACT FROM AUTHOR]

There is a well-known classification of conjugacy classes of involutions in finite Coxeter groups, in terms of subsets of nodes of their Coxeter graphs. In many cases, the product of an involution with the longest element w o is again an involution. We identify the conjugacy class of this product involution in terms of said classification. Communicated by Mandi Schaeffer Fry [ABSTRACT FROM AUTHOR]

In this paper, we determine all finite groups in which each element has order a power of a prime and have at most five non-central conjugacy classes with representatives of orders multiples of p, for each arbitrary prime number p. [ABSTRACT FROM AUTHOR]