• DR SUBHASH CHANDRA SINGH

The aim of this paper is to introduce a new representation of dihedral group Dn of degree n as a group of residue classes and study its properties. We find the (N,M)-th Commutativity degree P_N^M (D_n)for all positive integers N,M and n. P_N^M (D_n )is the probability of a random pair (x,y) of D_n×D_nso that . Let D_n^K={a^K ├|a∈ ┤ D_n }for a positive integer K. Further We find the relative (N,M)-th commutativity degree for all positive integers N,M and n. P_N^M 〖(D〗_n,D_n)is the probability that a random element of commutes with a random element ofD_n^M.Finally We find all subgroups, all normal subgroups, the center and the commutator subgroup of D_n.

Dihedral group Residue Classes, N- th commutativity degree, relative commutativity degree, Equivalence relation

"SOME PROPERTIES OF DIHEDRAL GROUP REPRESENTING AS A GROUP OF RESIDUE CLASSES", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.9, Issue 10, page no.d459-d485, October-2022, Available :http://www.jetir.org/papers/JETIR2210372.pdf

"SOME PROPERTIES OF DIHEDRAL GROUP REPRESENTING AS A GROUP OF RESIDUE CLASSES", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.9, Issue 10, page no. ppd459-d485, October-2022, Available at : http://www.jetir.org/papers/JETIR2210372.pdf