Preliminaries

Here we record some basic definitions and elementary results about group, rings, and varieties of algebraic systems. We assume a basic knowledge in undergraduate algebra. In particular, in group theory, the reader is supposed to be familiar with definitions and basic properties of subgroups, cosets, cyclic subgroups, direct products, the structure of finite abelian groups, normal subgroups, the Homomorphism Theorems. The Sylow Theorems may be referred to occasionally, but they are not used in the proofs of the main results of the book, which are all about finite p -groups. Some familiarity with rings and modules is assumed, although many of the definitions are briefly reproduced. We shall often use exponent notation for images under mappings, that is, a φ or A φ for φ( a ) or φ( A ) respectively, and sometimes also the right operator notation, a φ for φ( a ), say. The identity mapping of a set M will be denoted by 1 M . Standard notation ℕ, ℤ, ℚ, ℝ, ℂ is fixed for the sets of natural numbers, integers, rational, real and complex numbers, respectively. We shall say that a value is ( a, b )-bounded, say, if there is a function depending only on a and b , such that the value does not exceed this function. Groups Some basic definitions. The sets ℤ, ℚ, ℝ, ℂ are groups with respect to addition. The set ℤ/ n ℤ of residues modulo n is a group with respect to addition mod n . Every vector space is a group with respect to addition.