Unit groups of finite group algebras of Abelian groups of order at most 16

Abstract

In this paper, we establish the structure of the unit group of the group algebra [Formula: see text] where [Formula: see text] is an abelian group of order at most 16 and [Formula: see text] is a finite field of characteristic [Formula: see text] with [Formula: see text] elements.

Unit groups of group algebras of Abelian groups of order 32

“…If q ≡ −7 mod 32, then T = {1, 9, 17, 25} mod 32. Thus, p-regular Fconjugacy classes are {1}, {a, a 9 , a 17 , a 25 }, {a 2 , a 18 }, {a 3 , a 11 , a 19 , a 27 }, {a 4 }, {a 6 , a 22 }, {a 5 , a 13 , a 21 , a 29 }, {a 7 , a 15 , a 23 , a 31 }, {a 8 }, {a 10 , a 26 }, {a 12 }, {a 16 }, {a 14 , a 30 }, {a 20 }, {a 24 }, {a 28 } and c = 16. Hence…”

“…Our problem is based on the Witt-Berman theorem [6, Ch.17, Theorem 5.3], which states that the number of non-isomorphic simple F G-modules is equal to the number of F -conjugacy classes of p-regular elements of G. Problem of finding unit groups of group algebras generated a considerable interest in recent decade and can be easily seen in [5,7,8,10,[13][14][15]. Recently in [1,12], Sahai and Ansari have characterized the unit groups of group algebras of groups of orders 16 and 20. Let G be a group of order 32, we have seven non-isomorphic abelian groups C 32 ,…”

Unit Groups of Group Algebras of Abelian Groups of order 32

“…Thus p-regular F -conjugacy classes are {1}, {a, a 5 , a 9 , a 13 , a 17 , a 21 , a 25 , a 29 }, {a 2 , a 10 , a 18 , a 26 }, {a 4 , a 20 }, {a 3 , a 7 , a 11 , a 15 , a 19 , a 23 , a 27 , a 31 }, {a 8 }, {a 6 , a 14 , a 22 , a 30 }, {a 16 }, {a 24 }, {a 12 , a 28 } and c=10. Hence…”

“…Our problem is based on the Witt-Berman theorem [6, Ch.17, Theorem 5.3], which states that the number of non-isomorphic simple F G-modules is equal to the number of F -conjugacy classes of p-regular elements of G. Problem of finding unit groups of group algebras generated a considerable interest in recent decade and can be easily seen in [2,5,7,8,10,[13][14][15]. Recently in [1,12], Sahai and Ansari have characterized the unit groups of group algebras for the abelian groups of orders up to 20. Let G be a group of order 32, we have seven non-isomorphic abelian groups C 32 , C 16 × C 2 , C 8 ×C 4 , C 8 ×C 2 ×C 2 , C 4 ×C 4 ×C 2 , C 4 ×C 3 2 and C 5 2 .…”

The structure of the unit group of the group algebra F(C₃ × D₁₀)

“…Many authors [1,4,5,7,[9][10][11][12] have studied the structure of ( ) for a finite group and for a finite field . The structure of ( ( 3 × 10 )) for = 3 is…”