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Unit groups of group algebras of some small groups

Gaohua TangYangjiang WeiYuanlin Li — 2014

Czechoslovak Mathematical Journal

Let F G be a group algebra of a group G over a field F and 𝒰 ( F G ) the unit group of F G . It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group G with order 21 over any finite field of characteristic 3 is established. We also characterize the structure of the unit group of F A 4 over any finite field of characteristic 3 and the structure of...

Structure of cubic mapping graphs for the ring of Gaussian integers modulo n

Yangjiang WeiJizhu NanGaohua Tang — 2012

Czechoslovak Mathematical Journal

Let n [ i ] be the ring of Gaussian integers modulo n . We construct for n [ i ] a cubic mapping graph Γ ( n ) whose vertex set is all the elements of n [ i ] and for which there is a directed edge from a n [ i ] to b n [ i ] if b = a 3 . This article investigates in detail the structure of Γ ( n ) . We give suffcient and necessary conditions for the existence of cycles with length t . The number of t -cycles in Γ 1 ( n ) is obtained and we also examine when a vertex lies on a t -cycle of Γ 2 ( n ) , where Γ 1 ( n ) is induced by all the units of n [ i ] while Γ 2 ( n ) is induced by all the...

The fundamental constituents of iteration digraphs of finite commutative rings

Jizhu NanYangjiang WeiGaohua Tang — 2014

Czechoslovak Mathematical Journal

For a finite commutative ring R and a positive integer k 2 , we construct an iteration digraph G ( R , k ) whose vertex set is R and for which there is a directed edge from a R to b R if b = a k . Let R = R 1 ... R s , where s > 1 and R i is a finite commutative local ring for i { 1 , ... , s } . Let N be a subset of { R 1 , , R s } (it is possible that N is the empty set ). We define the fundamental constituents G N * ( R , k ) of G ( R , k ) induced by the vertices which are of the form { ( a 1 , , a s ) R : a i D ( R i ) if R i N , otherwise a i U ( R i ) , i = 1 , ... , s } , where U ( R ) denotes the unit group of R and D ( R ) denotes the zero-divisor set of R . We investigate...

The cubic mapping graph for the ring of Gaussian integers modulo n

Yangjiang WeiJizhu NanGaohua Tang — 2011

Czechoslovak Mathematical Journal

The article studies the cubic mapping graph Γ ( n ) of n [ i ] , the ring of Gaussian integers modulo n . For each positive integer n > 1 , the number of fixed points and the in-degree of the elements 1 ¯ and 0 ¯ in Γ ( n ) are found. Moreover, complete characterizations in terms of n are given in which Γ 2 ( n ) is semiregular, where Γ 2 ( n ) is induced by all the zero-divisors of n [ i ] .

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