This paper is the fifth part of the publication ``Selected chapters from algebra'', the first four having been published in previous issues of the Teaching of Mathematics, Vol. I (1998), 1--22, Vol. II, 1 (1999), 1--30, Vol. II, 2 (1999), 65--80, Vol. III, 1 (2000), 15--40, and Vol. III, 2 (2000), 63--82.