Groups A group is a nonempty set G equipped with a binary operation * that satisfies the following axioms: 1. Closure:If a is in G and b is in G then a * b is in G. 2. Associativity: a*(b*c) = (a*b)*c for all a, b, c in G. 3. There is an element e in G called the identity element such that a*e = a = e*a for every a in G. 4. For each a in G, there is an element d in G called the inverse of the element a such that a*d = d*a = e. A group is said to be abelian if it also satisfies this axiom: 5. Commutativity: a*b = b*a for all a, b in G. The order of a finite group is the number of elements in the group. The order of an element is the least power of the element which = the identity. S_3 is the symmetric group on 3 symbols. It has order 6. The elements of S_3 are {(1),(1,2),(1,3)(2,3),(1,2,3),(1,3,2)}. The subset of S_3, { (1), (1,2,3), (1,3,2) ) also is a group. D_4 is the group of translations of a square onto itself. D_4 consists of 4 rotations and 4 flips. D_4 = {r_0,r_1,r_2,r_3,f_0,f_1,f_2,f_3} Where r_0 = rotation of zero degrees counterclockwise. r_1 = rotation of 90 degrees counterclockwise. r_2 = rotation of 180 degrees counterclockwise. r_3 = rotation of 270 degrees counterclockwise. f_0 = flip around x-axis.( Called d in the textbook) f_1 = flip around y=x axis.(Called h in the textbook) f_2 = flip around y-axis.(Called t in the textbook) f_3 = flip around y=-x axis.(called v in the textbook) The set {1, i, -1, -i } is a group under the operation of complex number multiplication. The positive rational numbers form a group under real number multiplication. The last two examples are special cases of: In a ring with unity, the units of the ring form a group under the ring multiplication. Z_n is the ring of integers mod n. Z_n also refers to the additive group of this ring. U_n is the group of units of Z_n. GL(m,R) is the group of units in the ring of m by m matrices over the ring R. An m by m matrix is a unit if and only if its determinant is nonzero. Z_n has n elements. The order of Z_n is n. D_4 has 8 elements. The order of D_4 is 8. S_3 has 6 elements. The order of S_3 is 6. S_4 has 24 elements. The order of S_4 is 24. S_5 has 120 elements. The order of S_5 is 120. S_n has n! elements. The order of S_n is n!. If G is a group, and b is an element in G, the the set H = { b^k such that k is an integer} is a subgroup of G. Indeed, if G is only a semi-group then H = { b^k such that k is an integer} is a subset of G which is a group under the semi-group operation. Note: a semi-group requires only the closure and associativity axioms. U_3 has 2 elements. The order of U_3 is 2. U_4 has 2 elements. The order of U_4 is 2. U_5 has 4 elements. The order of U_5 is 4. U_6 has 2 elements. The order of U_6 is 2. U_7 has 6 elements. The order of U_7 is 6. U_8 has 4 elements. The order of U_8 is 4. U_9 has 6 elements. The order of U_9 is 6. U_10 has 4 elements. The order of U_10 is 4. U_11 has 10 elements. The order of U_11 is 10. U_12 has 4 elements. The order of U_12 is 4. U_n has as number of elements the number of positive integers < n that are relative prime to n. Define phi(n) = number of positive integers less than n that are relative prime to n. Order(U_n) = phi(n). Let f be a 1-1 map of the elements of a group onto itself. Define a new group operation @ by A @ B = f^-1 ( f(A) * f(B) ). Then G is also a group under the new operation @. The bi-jective functions of a set onto itself form a group under the operation of composition of functions. SL(m,R) is the group of m by m matrices over the ring R such that their determinant = 1. In the operation table for a finite group, each element appears exactly once in each row and column. The set of linear functions over the reals form a group under the operation composition of functions.