This paper is the seventh (and last) part of the publication ``Selected chapters from algebra'', the first six 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, Vol\. III, 2 (2000), 63--82, Vol\. IV, 1 (2001), 1--34 and Vol\. IV, 2 (2001), 71--104.