Session 9, Part C:
Groups and Fields

The development of mathematical structures in the 19th century included the expansion of number systems into larger sets. Mathematicians studied the properties of these sets, and their operations, so that equations in any system would have roots in the new expanded system. These explorations led Galois to develop the concept of what is now known as a group. The definition of a group is as follows:
Note: Optional Problems

With a set S and an operation *, S is a group under * if all four of these are satisfied (Note that one this page, the "*" represents any binary operation, and not only multiplication as elsewhere in this Part.):

 1 S is closed under * 2 S is associative under * 3 S has an identity element under * 4 Every element of S has an inverse

S can be any set of objects, and * can be any binary operation. Here is more information about each of these. (Note that in this Part, the "*" represents any binary operation, and not only multiplication as in previous Parts.)

S is closed under * if the operation always produces outputs within S. The set of whole numbers is closed under addition and multiplication -- add two whole numbers, and the answer is always a whole number. The set of whole numbers is not closed under subtraction or division, however, since the output is not always a whole number.

Problem C20

Which of the following sets are closed under addition? Which are closed under multiplication? Which are closed under division?

 a. Positive whole numbers: {1, 2, 3, 4, ...} b. Full set of integers: {..., -3, -2, -1, 0, 1, 2, 3, ...} c. All real numbers on the number line d. Positive real numbers e. Positive odd integers: {1, 3, 5, 7, ...}

 S is associative if, for elements a, b, and c in S, (a * b) * c = a * (b * c). This property is especially useful in solving equations. S has an identity element "e" if, for every element a in S, a * e = a and e * a = a. Then, e is the identity. Zero is the identity for addition, and 1 is the identity for multiplication. S has inverses if, for every element a in S, there is an element b in S for which a * b = e, the identity. The inverse of 3 under addition is -3, and the inverse of 3 under multiplication is 1/3. If a set S has no identity under *, it cannot have inverses.

Problem C21

Which of these sets and operations have an identity element? Which have inverses?

 a. Positive whole numbers, under addition b. Positive whole numbers, under multiplication c. Full set of integers, under addition d. Full set of integers, under multiplication e. Full set of integers, under division

All mathematical groups share these four fundamental properties. Many groups are also commutative if a * b = b * a for any elements in S.

Another related concept is a mathematical field. If a set S has two operations (usually + and x) defined on it, it is a field if all of these properties hold:

 1 (S, +) is a commutative group 2 (S, x) is a commutative group when the identity of the first operation is removed 3 The first operation, +, is distributive over the second operation, x. This means that for any elements a, b, and c in S, it must be true that a * (b + c) = (a * b) + (a * c).

This is a total of 11 properties. While it is rare for a set and two operations to be a field, many mathematical properties of one field translate directly to any other field.

The real numbers are a field under addition and multiplication.

Problem C22

Which of these are fields under the operations of addition and multiplication?

 a. Positive real numbers b. Rational numbers (integers and fractions) c. Even numbers: {..., -4, -2, 0, 2, 4, ...} d. Mod 4 e. Mod 5

 Session 9: Index | Notes | Solutions | Video

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