On the localization genus of a direct product of metacyclic groups
Peter J. Witbooi · Journal of Group Theory · 2009
For a relatively prime pair of natural numbers n, u , let G ( n ; u ) = 〈 a , b | a n = 1, bab –1 = a u 〉. For groups H of the type G ( n ; u ) × G ( m ; u ) we determine the set χ( H ) of all isomorphism classes of groups K with the property that K × ℤ ≅ H × ℤ. The set χ( H ) carries a certain group structure which coincides with the Hilton–Mislin genus group structure when H is nilpotent. We compute the group χ( H ).