Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.10, 2014 116 Some Results on the Group of Lower Unitriangular Matrices L(3,ℤ p ) Asmaa Abd Aswhad , Niran Sabah Jasim , Ahmed Rasim Hameed Department of Mathematics, College of Education for pure Science/ Ibn-Al-Haitham, University of Baghdad Abstract The main objective of this paper is to find the order and its exponent, the general form of all conjugacy classes, Artin characters table and Artin exponent for the group of lower unitriangular matrices L(3,ℤ p ), where p is prime number. Key Words: Artin character, Artin exponent, cyclic subgroup, group of unitriangular matrices. Introduction The group of invertible n  n matrices over a field F denoted by GL(n,F). Let G be a finite group, all characters of G induced from a principal character of cyclic subgroups of G are called Artin characters of G. Artin induction theorem [1] states that any rational valued character of G is a rational linear combination of the induced principal character of its cyclic subgroups. Lam [5] proved a sharp form of Artin's theorem, he determined the least positive integer A(G) such that A(G) is an integral linear combination of Artin character, for any rational valued character  of G, and he called A(G) the Artin exponent of G and studied it extensively for many groups. In this paper we consider the group of lower unitriangular matrices L(3,ℤ p ) and we found that the order of this group is p 3 as in theorem (2.2) and its exponent is p in theorem (2.4). Furthermore we found forms of all conjugace classes in theorem (2.8), the Artin character in theorem (2.12) and finally from the principal character of its cyclic subgroups we found the Artin exponent of this group and denoted by A(L(3, ℤ p )) which is equal to p 2 in theorem (2.13). §.1 Preliminaries In this section, we recall some definitions, theorems and proposition which we needed in the next section. Definition 1.1 : [4] A rational valued character  of G is a character whose valued are in Z, that is (x)Z, for all xG. Definition 1.2 : [2] Let H be a subgroup of a group G, and  be a class function of H. Then ↑ G , the induced class function on G, is given by 1 x G 1 (g) (xgx ) H       , where ˚ is defined by ˚(h) = (h) if h  H and ˚(y) = 0 if y  H. Observe that ↑ G is a class function on G and ↑ G (1) = [G:H] (1).