1. Groups Definition A group  G,  is a set G, closed under a binary operation , such that the following axioms are satisfied: 1)Associativity of  : For all a, b, c  G, we have (a  b)  c = a  (b  c). 2) Identity element e for  : There is an element e in G such that for all x  G, e  x = x  e = x 3) Inverse a’ of a: For each a  G, there is an element a’ in G such that a  a’ = a’  a = e.

2. Examples Definition A group G is abelian if its binary operation is commutative. Example: The set Z + under addition is not a group. since there is no identity element for + in Z +. The set of all nonnegative integers under addition is not a group since there is no inverse for 2. The set Z, Q, R, and C under addition are abelian groups. The set Z + under multiplication is not a group since there is no inverse of 3.

3. Examples The sets Q + and R +, Q *, R *, C * under multiplication are abelian groups. Example: Let  be defined on Q + by a  b=ab/2. Determine Q + under such a binary operation  is a group. It is a group since it satisfies the three properties of a group: (1)(a  b)  c=  (ab/2)  c=(abc)/4, and a  (b  c)=  a  (bc/2)=(abc)/4. Thus  is associative. (2)a  e=ae/2=a implies e=2. We have 2  a=a  2=a for all a  Q +. So 2 is an identity element for . (3)a  a’=aa’/2=e=2 implies a’=4/a. We have a  4/a=4/a  a=2. So a’=4/a is an inverse for a. Hence Q + with the operation  is a group. #

4. Elementary Properties of Groups Theorem 4.15 If G is a group with binary operation , then the left and right cancellation laws hold in G. that is, a  b=a  c implies b=c, and b  a=c  a implies b=c for all a, b, c  G. Proof: Suppose a * b = a * c. Then there exists an inverse of a’ to a. Apply this inverse on the left, a’ * (a * b) = a’ *(a * c) By the associatively law, (a’ * a ) * b = (a’ * a) * c Since a’ is the inverse of a, a’ * a =e, we have e * b = e * c By the definition of e, b = c Similarly for the right cancellation. #

5. Theorem 4.16 If G is a group with binary operation , and if a and b are any elements of G, then the linear equations a  x=b and y  a=b have unique solutions x and y in G. Proof: First we show the existence of at least one solution by just computing that a’  b is a solution of a  x=b. Note that a * (a’ * b) = (a *a’) * b, associative law, =e * b, definition of a’, =b, property of e. Thus x= a’  b is a solution a  x=b. In a similar fashion, y=b  a’ is a solution of y  a=b. To show uniqueness of y, we assume that we have two solutions, y 1 and y 2, so that y 1  a=b and y 2  a=b. Then y 1  a=y 2  a, and by Theorem 4.15, y 1 =y 2. The uniqueness of x follows similarly. #

6. Theorem 4.17 In a group G with binary operation , there is only one element e in G such that e  x = x  e = x for all x  G. Likewise for each a  G, there is only one element a’ in G such that a’  a = a  a’ = e In summary, the identity element and inverse of each element are unique in a group. Proof: We’ve shown the uniqueness of an identity element for any binary structure in section 3.

7. Uniqueness of an inverse Suppose that a  G has an inverses a’ and a’’ so that a’  a = a  a’ = e and a’’  a = a  a’’ = e. Then a  a’’= a  a’ = e And, by Theorem 4. 15, a’’=a’ So the inverse of a in a group is unique. #

8. Corollary Let G be a group. For all a, b  G, we have (a  b)’ = b’  a’. Proof: in a group G, we have (a  b)  (b’  a’) = a  (b  b’)  a’ = (a  e)  a’= a  a’=e.By theorem 4.17, b’  a’ is the unique inverse of a  b. That is, (a  b)’ = b’  a’. #

9. Group Table Every group table is a Latin square; that is, each element of the group appears exactly once in each row and each column. Proof: On the contrary, suppose x appears in a row labeled with a twice. Say x=a  b and x=a  c. Then cancellation gives b=c. This contradicts the fact that we use distinct elements to label the columns. #

10. Finite Groups One-element Group {e} with the binary operation  defined by e  e=e Two-element Group Example: {e, a}, try to find a table for a binary operation  on {e, a} that gives a group structure on {e, a}.  e a e e a a a x Since every element can occur exactly once in each row and each column, we have x =e.

11. Three-element group Example: {e, a, b}, try to find a table for a binary operation  on {e, a, b} that gives a group structure on {e, a, b}.  e a b e e a b a a x y b b z w Since every element can occur exactly once in each row and each column, we have x=b, y=e, z= e, w=a. Note: There is only one group of three elements, up to isomorphism.