COMMUTATIVITY DEGREES AND RELATED INVARIANTS OF SOME FINITE NILPOTENT GROUPS FADILA NORMAHIA BINTI ABD MANAF UNIVERSITI TEKNOLOGI MALAYSIA

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1 COMMUTATIVITY DEGREES AND RELATED INVARIANTS OF SOME FINITE NILPOTENT GROUPS FADILA NORMAHIA BINTI ABD MANAF UNIVERSITI TEKNOLOGI MALAYSIA

2 COMMUTATIVITY DEGREES AND RELATED INVARIANTS OF SOME FINITE NILPOTENT GROUPS FADILA NORMAHIA BINTI ABD MANAF A thesis submitted in fulfilment of the requirements for the award of the degree of Doctor of Philosophy (Mathematics) Faculty of Science Universiti Teknologi Malaysia MARCH 2013

3 iii To My Mom and Dad To My Beloved Husband To My Family

4 iv ACKNOWLEDGEMENTS In the name of Allah, the most gracious and merciful. He who has given me strength, courage and calm in completing my research. First of all, I would love to express my greatest gratitude to my beloved supervisor, Assoc. Prof. Dr. Nor Haniza Sarmin for her guidance, motivation, patience and incredible help throughout the process of doing this thesis until the end. I am also thankful to both of my co-supervisors, Dr Nor Muhainiah Mohd Ali and Prof Dr Ahmad Erfanian for their helps and for sharing their expertise during my studies. I want to thank my co-author, Dr Francesco Russo from the Department of Electrical, Electronics and Telecommunications Engineering (DIEETCAM), University of Palermo, Italy for his assistance in completing some parts of this thesis. I am grateful for his direct supervision while I was at Piedimonte Matese, Italy, for a two-week attachment (15 29 April 2012). Thanks also to Behnaz Tolue and Mohammad Farrokhi from Ferdowsi University of Mashhad, Iran, for their helps in completing parts of my research. The discussions I had with them were the most invaluable ones I have ever experienced. My special thanks to my family, especially my parents, for their continuous support and patience. Last but not least, thanks to my beloved husband, Muhammad Hafiz Mohd Hamdan and my dearest friend, Hazzirah Izzati Mat Hassim who have helped me in many ways in the completion of this thesis. Thank you.

5 v ABSTRACT In this research, two-generator p-groups of nilpotency class two, which is referred to as G are considered. The commutativity degree of a finite group G, denoted as P G, is defined as the probability that a random element of the group G commutes with another random element in G. The main objective of this research is to derive the general formula for P G and its generalizations. This research starts by finding the formula for the number of conjugacy classes of G. Then the commutativity degree of each of these groups is determined by using the fact that the commutativity degree of a finite group G is equal to the number of conjugacy classes of G divided by the order of G. The commutativity degree can be generalized to the concepts of n-th commutativity degree, Pn G, which is defined as the probability that the n-th power of a random element commutes with another random element from the same group. Moreover, Pn G commutativity degree, can be extended to the relative n-th P H, G, which is the probability of commuting the n-th n power of a random element of H with an element of G, where H is a subgroup of G. In this research, the explicit formulas for Pn G and n, P H G are computed. Meanwhile, another generalization of the commutativity degree, which is called commutator degree and denoted by Pg G, is the probability that the commutator of two elements in G is equal to an element g in G. In this research, an effective character-free method is used for finding the exact formula for Pg G. ^ exterior degree of the wreath product of A and B, P A B are two finite abelian groups., Finally, the is found where A and B

6 vi ABSTRAK Dalam penyelidikan ini, kumpulan-p berpenjana-dua dengan kelas nilpoten dua, dirujuk sebagai G, dipertimbangkan. Darjah kekalisan tukar tertib bagi suatu kumpulan terhingga G, ditandakan sebagai P G, ditakrifkan sebagai kebarangkalian bahawa satu unsur dalam G yang dipilih secara rawak adalah kalis tukar tertib dengan unsur lain yang dipilih secara rawak dalam G. Objektif utama kajian ini adalah untuk mendapatkan rumus umum bagi P G P G. dan pengitlakan bagi Kajian ini dimulakan dengan mencari rumus untuk bilangan kelas kekonjugatan bagi G. Kemudian darjah kekalisan tukar tertib bagi kumpulankumpulan ini ditentukan dengan menggunakan fakta bahawa darjah kekalisan tukar tertib bagi suatu kumpulan terhingga G adalah sama dengan bilangan kelas kekonjugatan dibahagi dengan peringkat kumpulan. Darjah kekalisan tukar tertib boleh diitlakkan kepada konsep darjah kekalisan tukar tertib kali ke-n, Pn G, yang ditakrifkan sebagai kebarangkalian bahawa kuasa ke-n bagi suatu unsur yang dipilih secara rawak berkalis tukar tertib dengan unsur yang lain daripada kumpulan yang sama. Selain itu, Pn G tertib secara relatif, P H, G n boleh dilanjutkan kepada konsep darjah kekalisan tukar yang ditakrifkan sebagai kebarangkalian berkalisan tukar tertib kuasa ke-n bagi suatu unsur dalam H yang dipilih secara rawak dengan suatu unsur dalam G, dengan H adalah sub-kumpulan bagi G. Dalam kajian ini, rumus eksplisit untuk n P G dan P H, G n dikira. Sementara itu, pengitlakan yang lain bagi darjah kekalisan tukar tertib, yang dinamai darjah pengalis tukar tertib dan dilambangi Pg G, merupakan kebarangkalian bahawa penukar tertib bagi dua unsur dalam kumpulan G adalah sama dengan unsur g yang diberikan dalam kumpulan tersebut. Dalam penyelidikan ini, satu kaedah bebas-aksara yang berkesan telah digunakan untuk mencari rumus tepat bagi Pg G. ^ peluaran bagi hasil darab kalungan A dan B, P A B kumpulan abelan terhingga A dan B., Akhir sekali, darjah diperoleh bagi dua