Groups Group theory is an example of "abstract algebra" (what mathematicians mean when they say algebra, as opposed to what you learned in high school) Concerned with underlying structure in mathematical systems that shows up repeatedly in different contexts. A group is the simplest of the objects studied in abstract algebra. and consists of a triple (S, e, *) with the following description. 1. S is a set of elements (finite or infinite) 2. e is a distinguished element of S called the "identity" 3. * is an operation (functional relation) mapping pairs of elements of S to elements of S that satisfies the following properties * Identity property: For all x in S, x * e = e * x = x. * Existence of inverse: For all x in S, there exists a y such that x * y = y * x = e. y is called the inverse of x. * Associativity: for all x,y,z in S, (x * y) * z = x * (y * z). The operation is frequently written as multiplication x*y, xy, etc. In this case, the inverse of x is often written as x^-1, and the identity is written as 1. If the operation * is commutative, that is, if x * y = y * x for all x,y, then the group is called "abelian" (after Neils Henrik Abel, a 19th century Norwegian mathematician). In this case, the group operation is sometimes written as addition +, the inverse as -x, and the identity as 0. Examples of groups and non-groups * The integers with addition (group, identity 0, inverse -x, abelian) * The integers with multiplication (nongroup - no inverse) * The rationals with multiplication (nongroup - 0 has no inverse) * The non-zero rationals with multiplication (group, identity (1,1), inverse of (i,j) is (j,i), abelian. * The integers with subtraction (nongroup - associativity does not hold) * The even integers with addition (group, identity 0, inverse -x, abelian) * The set Z/m of integers mod m with addition (group, identity 0 mod m, inverse -x or m-x mod m, abelian) * The set Z/m of integers mod m with multiplication (nongroup, 0 and sometimes other elements have no inverse) * The set Z/m* of integers mod m relatively prime to m with multiplication (group, identity 1 mod m, inverse computable with Euclidean algorithm, abelian). * The set of vectors in R^n with vector addition (group, identity 0 vector, inverse -x, abelian) * The set of vectors in R^n with dot product (nongroup - result of operation is not a vector) * The set of vectors in R^3 with vector cross product (nongroup - no identity (hence no inverse)) * The set of permutations with composition (group, identity is no-op permutation, non-abelian) * The set of nxn matrices of real numbers with matrix multiplication (nongroup - inverses do not always exist) * The set of invertible nxn matrices of reals with matrix multiplication (group, non-abelian) * The set of subsets of a set S with set union as operation (nongroup - can't get both inverse and identity) * The set of bijections of a set S onto itself (group, non-abelian) Subgroups A subset H of a group G is said to be a subgroup of G if it is a group in its own right with the same operation. * H is a subgroup if it contains e, and is closed under * and inversion. * For any group G, the set of the identity {e} and the whole set G are (trivial) subgroups. * The set of even integers with addition is a subgroup of the integers with addition. * The set of multiples of 4 with addition is a subgroup of the evens with addition (and of the integers with addition) * For the group Z/4+ of integers mod 4 (with addition), the elements represented by 0 mod 4 and 2 mod 4 form a subgroup * For the group Z/5+ of integers mod $, the elements 0, 2 and 4 (mod 5) do NOT form a subgroup. * For the group Z/10+ the elements 0 and 5 (mod 10) form a subgroup. Also the elements 0,2,4,6, and 8 (mod 10) * There is the suggestion of a pattern here... Lagrange's Theorem * For a finite group G, the "order" |G| of G is the number of elements in G. If H is a subgroup of a finite group G, then the order of H divides the order of G (|G| is multiple of |H|). Proof: The basic idea behind the proof is to look at what happens to various element of G if we try multiplying them by the elements in H. For example, if x is in H, then the result of multiplying it by something in H had better be in H (because of closure) On the other hand, if x is not in H, then the result of multiplying it by something in H had better NOT be in H (if y and x * y are in H, then x * y * y^1 = x would have to be in H which is a contradiction) What's more, distinct elements of H have to go somewhere different when multiplied by x (xy = xz --> y = z by existence of x^-1). So for every x, we can generate a set of the same size as H, by multiplying it (on one side) by every element in H. The set xH = {xy: y is in H} is called a "left coset" of H. Now if we take some element z of the coset generated b y and multiply it by some element of w H, the result must be in the same coset. To see this, note that z = yx for y in H. So wz = wyx, and wy is in H, so wz is in the left coset of H generated by x. So taking this all together with the observation that every x in G must end up in some coset (multiply it by e) we see there is some collection of disjoint cosets, all of the same size that make up or "partition" G. So the number of cosets and their size, are factors of |G|. The size is the order of H, hence Lagranges theorem. The other factor, the number of cosets, is sometimes called the "index" of H in G, denoted [G:H] * For subgroups H, I of G with I contained in H, the indices satisfy [G:I] = [G:H] * [H:I]. Cyclic Subgroups If we pick some element x of a group G, then the set = {x^n: n in the set Z of integers} forms a subgroup called the "cyclic subgroup of G generated by x" * The smallest n such that x^n = e is called the "order" of x * For finite groups, this n is the number of elements in (the order of) the subgroup. This is easy to prove using the "laws" of exponents (also easily proved) and the fact that n is the least integer for which x^n = e. Lagrange's theorem provides a simple proof for Euler's theorem. (For x relatively prime to n, x^Phi(n) = 1 mod n) First note that Z/n^x - the set of integers mod n relatively prime to n is a group under multiplication, and that its order is Phi(n) From Lagrange's theorem, and the discussion on cyclic subgroups, we know that the order k of any element x of Z/n* must divide Phi(n) - the size of Z/n*. Thus x^Phi(n) = (x^k)^(Phi(n)/k) = e^(Phi(n)/k) = e = 1 mod n. * So what Euler theorem is really saying is that Phi(n) turns the crank enough times that all the generators in the group line up at their repeat points. (Sort of a lcm upper bound result). * In fact, we can define the "exponent" of a group as the smallest l such that x^l = e for all x in G. It is easy to see that l is the lcm of the orders of the elements of G * And Lagranges theorem says that l must divide the size of the group G (but not that l is equal to the size of G) Group Isomorphism Formalization of the concept that groups may be "essentially identical" except for the names of the elements. Two groups G and H are "isomorphic" iff there exists a bijection f mapping G to H such that if x, y, z in G satisfy x * y = z, then f(x) * f(y) = f(z); In particular f(e) = e. * For example, the even integers with addition are isomorphic to the integers with the bijection f(x) = 2x. * For finite groups, isomorphic groups must have the same order.