Journal of Algebra and Its Applications Vol. 12, No. 8 (2013) 1350059 (7 pages) c  World Scientific Publishing Company DOI: 10.1142/S021949881350059X ON THE UNITARY UNITS OF THE GROUP ALGEBRA F 2 mM 16 ZAHID RAZA Department of Mathematics National University of Computer and Emerging Sciences B-Block, Faisal Town, Lahore, Pakistan zahid.raza@nu.edu.pk MAQSOOD AHMAD Department of Mathematics COMSATS Institute of Information Technology Lahore, Pakistan mahmad@ciitlahore.edu.pk Received 24 January 2013 Accepted 14 March 2013 Published 21 June 2013 Communicated by L. Bokut In this note, we have given the center Z(V*(F 2 mM 16 )) of unitary units subgroup V*(F 2 mM 16 ) of group algebra F 2 mM 16 , where M 16 = 〈x, y | x 8 = y 2 =1, xy = yx 5 〉 is the Modular group of order 16 and F 2 m is any finite field of characteristic 2, with 2 m elements. The structure of the unitary unit subgroup V*(F 2 mM 16 ) of the group algebra F 2 mM 16 , is also described, see Theorem 3.1. Keywords : Group algebra; unitary unit group; modular-group; circulant matrices. Mathematics Subject Classification: 20C05, 16S34, 15A15, 15A33 1. Introduction Let G be any finite group and R be a ring then RG denotes the group ring of the group G over ring R. The sum and product of two elements of RG is defined as x +y = ∑ g∈G (α g + β g )g and xy = ∑ g∈G ( ∑ h∈G α g β h -1 g )g. Clearly, RG with respect to this addition and multiplication is ring and hence called group ring. If F is a field then FG is called group algebra of group G over field F. Now, the homomorphism A : FG → F given by A( ∑ g∈G α g g)= ∑ g∈G α g is called an augmentation mapping of FG. The set of all invertible elements, denoted by U (FG), is called unit group of 1350059-1 J. Algebra Appl. 2013.12. Downloaded from www.worldscientific.com by Mr Maqsood Ahmad on 10/31/13. For personal use only.