The Computation of the Nonabelian Tensor Product of Cyclic Groups of Order

Abstract

Katalah G dan H dua kumpulan yang bertindak ke atas satu sama lain dan masing-masing bertindak ke atasnya sendiri secara konjugasi, maka tindakan tersebut adalah serasi jika (gh)g' = g(h(g–1 g')) and for and . Keserasian tindakan adalah penting dalam penentuan hasil darab tensor tak abelan. Hasil darab tensor tak abelan, G⊗H telah diperkenalkan oleh Brown dan Loday pada tahun 1984. Hasil darab tensor tak abelan adalah kumpulan yang dijana oleh g⊗h dengan dua hubungan = dan g⊗hh' = (g⊗h)(hg⊗hh') bagi g,g'∈G dan h,h'∈Hdengan G dan H bertindak antara satu sama lain dalam tindakan yang serasi dan bertindak ke atas mereka sendiri dengan konjugasi. Pada tahun 1987, Brown dan rakan-rakan memberikan masalah terbuka dalam menentukan sama ada hasil darab tensor tak abelan dari dua kumpulan kitaran adalah juga kumpulan kitaran. Visscher pada tahun 1998 telah membuktikan bahawa hasil darab tensor tak abelan tidak semestinya kumpulan kitaran, tetapi kajian beliau hanya difokuskan kepada kumpulan kitaran berperingkat kuasa bagi dua dengan tindakannya berperingkat dua. Dalam makalah ini, keserasian dan hasil darab tensor tak abelan berperingkat p2 dengan tindakan berperingkat p ditentukan.(hg)h' = h(g(h–1 h'))g,g'∈Gh,h'∈Hgg'⊗h(gg'⊗gh)(g⊗h)

The compatible action graphs for finite cyclic 2-groups

“…Let G and H be groups which act on each other. These mutual actions are said to be compatible with each other and with the actions of G and H on themselves by conjugation if According to the presentation of the automorphism group of a cyclic 2-group, Mohamad (2012) given the necessary and sufficient conditions for a pair of actions that act compatibly with each other with specific order. If one of the actions journal.ump.edu.my/daam ◄ has order two, then the necessary and sufficient conditions for the other actions to act compatibly with each other are given in the following theorem.…”

“…In [1] defined the nonabelian tensor product as a pair of groups, G and H which acts on each other provided the actions satisfy the compatibility conditions: There are some researcher study on compatible action of cyclic groups, for example in [2] and in [3] studied on compatibility condition of finite cyclic groups of p-power order. Then, in [4] studied on the number of compatible pair of action for cyclic groups of 2-power order.…”

The Number of Compatible Pair of Actions for Cyclic Groups of 2-Power Order

“…A paper by Mohamad et. al (2012) gave the compatibility conditions and the non-abelian tensor product of the cyclic group of order 2 p with the actions of order p. While Sulaiman et al (2015), investigated some compatible pairs of nontrivial actions of order two and four for some cyclic groups of 2-power order.…”

“…Next, the number of auto-morphisms for two cyclic groups of 2-power order with a specific order done by Mohamad (2012) in given the following corollary. (2 , ) 2 .…”

“…Next, Mohamad (2012) gives the characterization of the compatible conditions when the actions have an order greater than two are stated as follow.…”