The attached paper is a revised version of a paper of mine published in 1975. It deals with the cultural evolution of social groups in human populations. It examines lineal groups composed people who trace descent from a common ancestor either through males or through females but not both. Lineal groups are often recognized in culturally defined groups such as lineages and clans. It shows how a random process can result in a relatively stable number of lineal groups in a population. The random processes that are considered are Markov processes. The text is available here (1) as a DVI file that can be read on any computer equipped with a TeX DVI reader, as a (2) LaTex file that can be read with the IBM Tech Explorer plugin, and as a PDF file that can be read with the Adobe Acrobat Reader 3.1, a widely available reader and plugin..