# Conjugacy classes and graphs of two-groups of nilpotency class two

Two elements a and b of a group are called conjugate if there exists an element g in the group such that gag??1 = b: The set of all conjugates in a group forms the conjugacy classes of the group. The main objective of this research is to determine the number and size of conjugacy classes for 2-gener...

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Main Author: Thesis English 2013 http://eprints.utm.my/id/eprint/43966/5/SheilaIlangovanPFS2013.pdf No Tags, Be the first to tag this record!
id my-utm-ep.43966 uketd_dc my-utm-ep.439662017-06-22T02:28:43Z Conjugacy classes and graphs of two-groups of nilpotency class two 2013-09 Ilangovan, Sheila QA Mathematics Two elements a and b of a group are called conjugate if there exists an element g in the group such that gag??1 = b: The set of all conjugates in a group forms the conjugacy classes of the group. The main objective of this research is to determine the number and size of conjugacy classes for 2-generator 2-groups of nilpotency class two. Suppose G is a 2-generator 2-group of class two which comprises of three types, namely Type 1, Type 2 and Type 3. The general formulas for the number of conjugacy classes of G are determined by using the base group and central extension method, respectively. It is found that for each type of the group G, the number of conjugacy classes consists of two general formulas. Moreover, the conjugacy class sizes are computed based on the order of the derived subgroup. The results are then applied into graph theory. The conjugacy class graph of G is proven as a complete graph. Consequently, some properties of the graph related to conjugacy classes of the group are found. This includes the number of connected components, diameter, the number of edges and the regularity of the graph. Furthermore, the clique number and chromatic number for groups of Type 1, 2 and 3 are shown to be identical. Besides, some properties of the graph related to commuting conjugacy classes of abelian and dihedral groups are introduced. 2013-09 Thesis http://eprints.utm.my/id/eprint/43966/ http://eprints.utm.my/id/eprint/43966/5/SheilaIlangovanPFS2013.pdf application/pdf en public phd doctoral Universiti Teknologi Malaysia, Faculty of Science Faculty of Science Universiti Teknologi Malaysia UTM Institutional Repository English QA Mathematics QA Mathematics Ilangovan, Sheila Conjugacy classes and graphs of two-groups of nilpotency class two Two elements a and b of a group are called conjugate if there exists an element g in the group such that gag??1 = b: The set of all conjugates in a group forms the conjugacy classes of the group. The main objective of this research is to determine the number and size of conjugacy classes for 2-generator 2-groups of nilpotency class two. Suppose G is a 2-generator 2-group of class two which comprises of three types, namely Type 1, Type 2 and Type 3. The general formulas for the number of conjugacy classes of G are determined by using the base group and central extension method, respectively. It is found that for each type of the group G, the number of conjugacy classes consists of two general formulas. Moreover, the conjugacy class sizes are computed based on the order of the derived subgroup. The results are then applied into graph theory. The conjugacy class graph of G is proven as a complete graph. Consequently, some properties of the graph related to conjugacy classes of the group are found. This includes the number of connected components, diameter, the number of edges and the regularity of the graph. Furthermore, the clique number and chromatic number for groups of Type 1, 2 and 3 are shown to be identical. Besides, some properties of the graph related to commuting conjugacy classes of abelian and dihedral groups are introduced. Thesis Doctor of Philosophy (PhD.) Doctorate Ilangovan, Sheila Ilangovan, Sheila Ilangovan, Sheila Conjugacy classes and graphs of two-groups of nilpotency class two Conjugacy classes and graphs of two-groups of nilpotency class two Conjugacy classes and graphs of two-groups of nilpotency class two Conjugacy classes and graphs of two-groups of nilpotency class two Conjugacy classes and graphs of two-groups of nilpotency class two conjugacy classes and graphs of two-groups of nilpotency class two Universiti Teknologi Malaysia, Faculty of Science Faculty of Science 2013 http://eprints.utm.my/id/eprint/43966/5/SheilaIlangovanPFS2013.pdf 1747817124904042496