In this paper, is a metacyclic 2-group of positive type of nilpotency of class at least three. Let be the set of all subsets of all commuting elements of of size two in the form of where and commute and each of order two. The probability that an element of a group fixes a set is considered as one of the extensions of the commutativity degree that can be obtained by some group actions on a set. In this paper, we compute the probability that an element of fixes a set in which acts on a set, by conjugation.
In this paper, Ω denotes the set of all subsets of commuting elements of G in the form of (a, b), where a and b commute, |a| = |b| = 2. The probability that a group element of G fixes a set is one of the generalizations of the commutativity degree that has been recently introduced. In this paper, the probability that an element of a group fixes a set for semi-dihedral groups and quasi-dihedral groups is found. The results obtained are then applied to graph theory, more precisely to the generalized conjugacy class graph.Mathematics Subject Classification: 20P05, 20B40, 97K30
In this paper, let G be a metacyclic 2-group of negative type of class two and class at least three. Let Ω be the set of all subsets of all commuting elements of size two in the form of (a, b), where a and b commute and |a| = |b| = 2. The orbit graph is a graph whose vertices are non-central orbits under group action of G on Ω. In this paper, the orbit graph of metacyclic 2-groups of negative type of nilpotency class two and class at least three is determined. Some graph properties are also provided.
An extension of the concept of commutativity degree named the probability that an element of a group fixes a set was introduced in 2013. Suppose is a metacyclic 5-group and is the set of all ordered pairs (x,y) in G*G such that lcm(|x|,|y|)=5, xy=yx and x is not equal to y. In this paper, the probability that an element of a metacyclic 5-group fixes the set is computed by using conjugation action.
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