groupname1 http://www.openmath.org/cd/group3.ocd 2006-06-01 2004-06-01 1 1 experimental Well known groups in group theory Written by Arjeh M. Cohen 2003-04-15 quaternion_group This symbol represents the quaternion group of order 8. The quaternion group is isomorphic to the group generated by a, b with presentation a^2 = b^2 = aba^(-1)b^(-1) and a^4 = 1. 4 2 -2 2 -1 -1 The center of Q has order 2. The derived subgroup of Q coincides with the center of Q. dihedral_group This symbol is a function with one argument, which should be a positive integer n. When applied to n it represents the dihedral group of order 2n. This is the group of all isometries (including reflections) of the regular n-gon in the plane. The dihedral group of order 2n is isomorphic to the group generated by a, b with presentation a^2 = b^n = 1 and a b a = b^(-1). 2 cyclic_group This symbol is a function with one argument, which should be a natural number n. When applied to n it represents the cyclic group of order n. generalized_quaternion_group This symbol is a function with one argument, which should be a positive integer. When applied to n it represents the generalized quaternion group of order 4n. This is the group with three generators a, b, and c and relations c = a^2 = b^n, c*a = a*c , b*c = c*b, a*b = b*a*c, and c^2 = 1.