ABSTRACT ALGEBRA 1, LECTURE NOTES 4: DEFINITIONS AND EXAMPLES OF MONOIDS AND GROUPS.

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1 ABSTRACT ALGEBRA 1, LECTURE NOTES 4: DEFINITIONS AND EXAMPLES OF MONOIDS AND GROUPS. ANDREW SALCH 1. Monoids. Definition 1.1. A monoid is a set M together with a function µ : M M M satisfying the following properties: Associativity: For all x, y, z M, we have an equality µ(µ(x, y), z) = µ(x, µ(y, z)). Existence of an identity element, aka unitality : There exists an element e M such that µ(e, x) = µ(x, e) = x for all x M. The element e is called an identity element for M or sometimes a unit element for M. A monoid M is called commutative if µ(x, y) = µ(y, x) for all x, y M. Example 1.2. (Examples of monoids.) The set of natural numbers N, with the function µ : N N N given by addition. That is, µ(x, y) = x + y. This monoid is commutative. The unit element is 0. The set of integers Z, with the function µ : Z Z Z given by addition. Again, this monoid is commutative. The unit element is 0. (Hopefully now you are getting the point that the function µ is best thought of as a binary operation, like addition or multiplication.) The set of natural numbers N, with the function µ given by multiplication. Again, this monoid is commutative. The unit element is 1. The set of positive integers, with the function µ given by multiplication. Again, this monoid is commutative. The unit element is 1. For any positive integer n, the set of all n-by-n matrices with entries in the real numbers, with the function µ given by matrix addition. Again, this monoid is commutative. The unit element is the zero matrix. For any positive integer n, the set of all n-by-n matrices with entries in the real numbers, with the function µ given by matrix multiplication. If n > 1, then this monoid isn t commutative, since matrix multiplication doesn t always satisfy the equation MN = NM! The unit element is the identity matrix (ones on the diagonal, zeros everywhere else). Here is a more analytic example: given an open subset U of the set of real numbers, the set of all differentiable functions U R forms a monoid under pointwise addition, that is, given two differentiable functions f, g : U R, we let f + g be the function f + g : U R given by ( f + g)(x) = f (x) + g(x) for all x U. (It is a standard exercise in introductory analysis to prove that this function f + g is differentiable if both f and g are.) This monoid is commutative. The unit element is the function sending every element of U to zero. Date: September

2 2 ANDREW SALCH Let S be a set. The set of all functions S S is a monoid, with binary operation given by composition, that is, µ( f, g) = f g. This monoid is usually not commutative (specifically, this monoid fails to be commutative as long as S has more than two elements). The unit element is the identity function id S. Example 1.3. (Non-examples of monoids.) The set of positive integers, under addition, is not a monoid, since although addition is associative, it does not have a unit element in the positive integers (since zero is not positive). The set of integers 2, under multiplication, is not a monoid, since although multiplication is associative, it does not have a unit element in the integers 2 (since 1 is not 2). The set of 2-by-2 matrices with real entries, equipped with the binary operation µ(m, N) = MN NM, which is often written [M, N] (this is called the commutator product), is not a monoid, since the commutator product is not associative. For example, if we let [ ] 0 1 L = M = N = 1 0 [ it is an elementary exercise in matrix multiplication to check that ], [[L, M], N] = LMN LNM MNL + NML LMN MLN NLM + NML = [L, [M, N]]. Here is an analytic example: let A be the set of compactly supported continuous functions from R to R. (A function f : R R is called compactly supported if there exists some bounded interval [a, b] R such that f (x) = 0 unless x [a, b].) The convolution product of compactly supported continuous functions f, g : R R is defined as the function f g : R R given by the formula ( f g)(x) = f (y)g(x y)dy. The convolution product is associative (and commutative), but the set A, equipped with the convolution product, is not a monoid, because there is no unit element! (This isn t obvious, but it is true.) Proposition 1.4. Let M be a monoid. Then M has only one unit element. That is, if e, e are both unit elements in M, then e = e. Proof. If e, e are both unit elements in M, then by the definition of a unit element, we have e = µ(e, e ) = e. 2. Groups. Definition 2.1. A group is a monoid G with the property that, for every element g G, there exists an element g 1 G such that µ(g, g 1 ) = µ(g 1, g) = e. A group is called an abelian group if its underlying monoid is commutative. That is, a group G is abelian if µ(x, y) = µ(y, x) for all x, y G.

3 ABSTRACT ALGEBRA 1, LECTURE NOTES 4: DEFINITIONS AND EXAMPLES OF MONOIDS AND GROUPS. 3 The reason that Definition 2.1 makes sense is that a monoid has a unique unit element e, by Proposition 1.4; so we don t have to say something like A group is a monoid G with the property that, for every element g G, there exists an element g 1 G such that µ(g, g 1 ) = µ(g 1, g) = e for some unit element e ; there s only one unit element for a given monoid, which makes the definition of a group simpler than it would otherwise be. Conventions 2.2. In most examples of groups, the binary operation µ is either some kind of generalized addition or some kind of generalized multiplication. It is very typical, when working with a particular group G, to write that you are going to use additive notation for G, and then to consistently write x + y instead of µ(x, y) when applying the group operation µ to elements x, y G. It is also very typical, when working with a particular group G, to write that you are going to use multiplicative notation for G, and then to consistently write xy instead of µ(x, y) when applying the group operation µ to elements x, y G. The convention is that, if you use additive notation, then the group G should be abelian; so to avoid confusion, you should use multiplicative notation for any group which is not abelian. But in general, for abelian groups, you have the choice of using either additive or multiplicative notation, and you should choose whichever notation seems to you to be the clearest and most intuitive notation for whatever group you are working with. Example 2.3. (Examples of groups.) The set with one element is a group in a unique way: if that element is called x, we have to let µ(x, x) = x. (It is very easy to check that the axioms for being a group are satisfied.) This group is called the trivial group. The monoid of integers Z under addition; the inverse of n is, of course, n. This is an abelian group. The monoid of nonzero rational numbers under multiplication; the inverse of n is, of course, 1/n. This is an abelian group. The monoid of nonzero real numbers under multiplication; the inverse of n is again 1/n. This is an abelian group. For any positive integer n, the monoid (under multiplication) of all n-by-n matrices with nonzero determinant with entries in the real numbers, with the function µ given by matrix multiplication. This uses the fact, which you learned in linear algebra, that a square matrix M with real entries has an inverse matrix M 1 if and only if the determinant of M is nonzero. This is a group, but if n > 1, it is non-abelian (because matrix multiplication isn t commutative). Given an open subset U of the set of real numbers, the set of all differentiable functions U R under pointwise addition is also a group: the inverse of a function f : U R is the function f given by ( f )(x) = f (x). This is an abelian group. Given a set S, the monoid of all bijective functions f : S S, under composition, is a group: the inverse of a function f is the inverse, in the sense of composition of functions. (See lecture notes #3.) This is a group, but if S has more than two elements, it is non-abelian. (See Example 2.5.) Example 2.4. (Non-examples of groups.) The monoid of natural numbers N under addition is a monoid, but not a group, because it doesn t have inverses: 0 is the unit element, but the element 1, for example, doesn t have any natural number n such that 1 + n = 0. (Of course n ought to be 1 here; but 1 is an integer, but not a natural number.)

4 4 ANDREW SALCH The monoid of positive integers under multiplication is a monoid, but not a group, because it doesn t have inverses: 1 is the unit element, but the element 2, for example, doesn t have any positive integer n such that 2 n = 1. (Of course n ought to be 1/2 here; but 1/2 is a rational number, but not a positive integer.) The set Q of all rational numbers under multiplication is a monoid but not a group, again because it doesn t have inverses: 1 is the unit element, but the element 0 doesn t have any rational number n such that 0 n = 1. (However, 0 is the only element in the rational numbers that doesn t have a multiplicative inverse; if we exclude zero and instead consider the set of all nonzero rational numbers under multiplication, then that is a group.) For any positive integer n, the monoid of all n-by-n matrices with entries in the real numbers, under matrix multiplication. This is a monoid, but not a group, since the matrices with determinant zero do not have inverses. Let S be a set. The set of all functions S S under composition is a monoid, but if S has more than one element, it is not a group, since any function S S which fails to be bijective will also fail to have an inverse. Example 2.5. Let n be a natural number, and let S be any set with exactly n elements. Then the group of bijective functions S S, under composition, is often called the symmetric group on n letters, and written S n or Σ n. There is a very useful notation for describing elements in symmetric groups, and it works like this: suppose we choose, as our set with n elements, simply the set of integers {1, 2,..., n}. Given some integers a 1,..., a b between 1 and n, we often write (a 1 a 2... a n ) for the bijection {1,..., n} {1,..., n} which sends a 1 to a 2, sends a 2 to a 3,..., sends a n 1 to a n, and sends a n to a 1. We often write (a 1 a 2... a n )(b 1 b 2... b m ) to mean the composite function (a 1 a 2... a n ) (b 1 b 2... b m ). The symmetric group on 1 letter, Σ 1, is the trivial group: there is only one function from a set with one element to a set with one element, so Σ 1 has only one element. The symmetric group on two letters, Σ 2, has two elements, the identity function e, and the function (12) which swaps the two elements. We have e(12) = (12)e = (12), and if we apply (12) twice, we get (12)(12) = e. It is convenient to represent this by a multiplication table: e (12) (12) e The symmetric group on three letters, Σ 3, has six elements: e, (12), (13), (23), (123), and (132). The group operation is given by the multiplication table: e (12) (13) (23) (123) (132) (12) e (132) (231) (23) (13) (13) (123) e (213) (12) (23) (23) (132) (123) e (13) (12) (123) (13) (23) (12) (132) e (132) (23) (12) (13) e (123) The way to read a multiplication table like this is as follows: to know the product xy, you look up the entry in row x and column y. For example, (12)(13) is the function that sends 1 to 3, sends 3 to 2, and sends 2 to 1, i.e., it is the function (132); so the entry in the multiplication table in row (12) and column (13) is (132). Example 2.6. Let n be a positive integer. Then there exists an abelian group with n elements called the cyclic group with n elements, and written C n or Z/nZ, defined as follows: let Z/nZ = {0, 1, 2,..., n 1}, with group operation given by addition modulo n. (It is easy

5 ABSTRACT ALGEBRA 1, LECTURE NOTES 4: DEFINITIONS AND EXAMPLES OF MONOIDS AND GROUPS. 5 to check that this indeed satisfies the associativity, unitality, inverses, and commutativity axioms, hence is a group.) For example, in the group Z/5Z, we have the multiplication table (although really it s an addition table ):