An Elementary Introduction to Groups and Representations (original) (raw)

The set of complex numbers with absolute value one (i.e., of the form e iθ) forms a group under complex multiplication. This group is commutative. This group is the unit circle, denoted S 1. 2.7. Invertible matrices. For each positive integer n, the set of all n × n invertible matrices with real entries forms a group with respect to the operation of matrix multiplication. This group in non-commutative, for n ≥ 2. We check closure: the product of two invertible matrices is invertible, since (AB) −1 = B −1 A −1. Matrix multiplication is associative; the identity matrix (with ones down the diagonal, and zeros elsewhere) is the identity element; by definition, an invertible matrix has an inverse. Simple examples show that the group is noncommutative, except in the trivial case n = 1. (See Exercise 8.) This group is called the general linear group (over the reals), and is denoted GL(n; R). 2.8. Symmetric group (permutation group). The set of one-to-one, onto maps of the set {1, 2, • • • n} to itself forms a group under the operation of composition. This group is non-commutative for n ≥ 3. We check closure: the composition of two one-to-one, onto maps is again oneto-one and onto. Composition of functions is associative; the identity map (which sends 1 to 1, 2 to 2, etc.) is the identity element; a one-to-one, onto map has an inverse. Simple examples show that the group is non-commutative, as long as n is at least 3. (See Exercise 10.) This group is called the symmetric group, and is denoted S n. A one-to-one, onto map of {1, 2, • • • n} is a permutation, and so S n is also called the permutation group. The group S n has n! elements. 2.9. Integers mod n. The set {0, 1, • • • n − 1} forms a group under the operation of addition mod n. This group is commutative. Explicitly, the group operation is the following. Consider a, b ∈ {0, 1 • • • n − 1}. If a + b < n, then a + b mod n = a + b, if a + b ≥ n, then a + b mod n = a + b − n. (Since a and b are less than n, a+b−n is less than n; thus we have closure.) To show associativity, note that both (a+b mod n)+c mod n and a+(b+c mod n) mod n are equal to a + b + c, minus some multiple of n, and hence differ by a multiple of n. But since both are in the set {0, 1, • • • n − 1}, the only possible multiple on n is zero. Zero is still the identity for addition mod n. The inverse of an element a ∈ {0, 1, • • • n − 1} is n − a. (Exercise: check that n − a is in {0, 1, • • • n − 1}, and that a + (n − a) mod n = 0.) The group is commutative because ordinary addition is commutative. This group is referred to as "Z mod n," and is denoted Z n. 3. Subgroups, the Center, and Direct Products Definition 1.7. A subgroup of a group G is a subset H of G with the following properties: 1. The identity is an element of H. 2. If h ∈ H, then h −1 ∈ H. 3. If h 1 , h 2 ∈ H, then h 1 h 2 ∈ H .