Section 1.3. Groups 17 addition, and subtraction are, respectively, the functions (x, y) → xy, (x, y) → x+y, and (x, y) → x − y. The examples of composition and subtraction show why we want ordered pairs, for x ∗ y and y ∗ x may be distinct. In constructing a binary operation on a set G, we must check, of course, that x, y ∈ G implies x ∗ y ∈ G if this is so, we say that G is closed under ∗. For example, if G is the set of all odd integers, then G is closed under multiplication (for xy is odd when both x and y are), but G is not closed under addition. As any function, a binary operation is well-defined when stated explicitly, this is usually called the law of substitution: if x = x and y = y , then x ∗ y = x ∗ y . Definition. A group is a set G equipped with a binary operation ∗ such that (i) the associative law 9 holds: for every x, y, z ∈ G, x ∗ (y ∗ z) = (x ∗ y) ∗ z (ii) there is an element e ∈ G, called the identity, with e ∗ x = x = x ∗ e for all x ∈ G (iii) every x ∈ G has an inverse: there is x ∈ G with x ∗ x = e = x ∗ x. The set SX of all permutations of a set X, with composition as binary operation and 1X = (1) as the identity, is a group (the symmetric group on X). Some of the equations in the definition of group are redundant. When verifying that a set with a binary operation is actually a group, it is obviously more econom- ical to check fewer equations (see Exercise 1.31 on page 27: a set G containing an element e and having an associative binary operation ∗ is a group if e ∗ x = x for all x ∈ G and, for every x ∈ G, there is x ∈ G with x ∗ x = e). We are now at the precise point where Algebra becomes abstract Algebra. In contrast to the concrete group Sn consisting of all the permutations of {1, 2,...,n}, we have passed to abstract groups whose elements are unspecified and whose prod- ucts are not explicitly computable (instead, multiplication is merely subject to certain rules). It will be seen that this approach is quite fruitful, for theorems now apply to many different groups, and it is more eﬃcient to prove theorems once for all instead of proving them anew for each group encountered. In addition to this obvious economy, it is often simpler to work with the “abstract” viewpoint even when dealing with a particular concrete group. Indeed, when we deal with ab- stract groups, we can more easily focus on the essential facts we need without being distracted by “noise.” For example, we will see that certain properties of Sn are simpler to treat without recognizing that the elements in question are permutations and the binary operation is composition (see Example 1.28). 9Not all binary operations are associative. For example, subtraction is not associative: if c = 0, then a − (b − c) = (a − b) − c, and so the notation a − b − c is ambiguous. The cross product of two vectors in R3 is another example of a nonassociative operation.

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