Group Theory: Definition and Examples, Study notes of Introduction to Sociology

Download Group Theory: Definition and Examples and more Study notes Introduction to Sociology in PDF only on Docsity! Background information about #43. (Not necessary to do the problem.) We will go into more detail later in the course. i.e. means ‘that is’ + means defined to be equal to N + {1, 2, 3, . . .} the set of natural numbers (i.e., positive integers) The following definition is on p. 244 (§8.3) of Fletcher and Patty. A group is a set G with a binary operation · which assigns to each a pair of elements a and b of G an element a ·b of G, where · has the following properties: (a) (associative law) (a · b) · c = a · (b · c) for all a, b, c ∈ G. (b) (identity element) there exists an element e of G such that a · e = e · a = a for every a ∈ G. (c) (inverse of an element) for each a ∈ G there exists x ∈ G such that a · x = x · a = e. We denote the inverse of a by a−1. If a ∈ N, then the powers of a are denoted by an + a · · · a (where a occurs n times). Examples of groups: 1. Z means the set of integers {. . . ,−2,−1, 0, 1, 2, . . .}. The integers with addition (Z, +) form a group. 2. R means the set of real numbers. The real numbers with addition (R, +) form a group. 3. The real numbers with multiplication (R, ·) is not a group since 0 does not have an inverse. 4. The nonzero real numbers with multiplication (R\ {0} , ·) is a group. We also define 1. a0 + e 2. for n a negative integer, an + ( a−n )−1 . Note that −n ∈ N so that a−n denotes a multiplied by itself with −n number of a’s and (a−n)−1 denotes the inverse of a−n. 1