ABELIAN GROUPS Also Known as Commutative Groups ABELIAN

ABELIAN GROUPS Also Known as Commutative Groups

ABELIAN GROUPS 1. In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers. They are named after early 19 th century mathematician Niels Henrik Abel. 2. Definition: If (G, ⋅) is a group where ⋅ is commutative for all a, b∈S, i. e. , a⋅b=b⋅a for all a, b∈G, then (G, ⋅) is called an Abelian Group. 3. The term "Commutative Group" means the same thing as "Abelian Group". 4. One such example of a group that is not abelian is the group (G, ⋅) where G is the set of 2× 2 matrices with real entries whose determinants are nonzero and ⋅ is defined to be matrix multiplication.

BASIC THEOREMS REGARDING ABELIAN GROUPS I. Theorem 1: Let (G, ⋅) be a group. If for all a, b∈G we have that (a⋅b)^2=a^2⋅b^2 then (G, ⋅) is an abelian group.

BASIC THEOREMS REGARDING ABELIAN GROUPS 2. Theorem 2: Let (G, ⋅) be a group. Then G is an abelian group if and only if for all a, b∈G we have that (a⋅b)− 1=a− 1⋅b− 1.

BASIC THEOREMS REGARDING ABELIAN GROUPS 3. Theorem 3: Let (G, ⋅) be a group. If for all a∈G we have that a=a− 1 then (G, ⋅) is an abelian group.

EXAMPLES OF ABELIAN GROUPS ØSome examples of Abelian groups are: The Integers under Addition, (Z, +) The Non-Zero Rational Numbers under Multiplication, (Q*, X) The Modular Integers under modular addition, (Zn, +) The U-groups, under modular multiplication, U(n) = {the set of integers less than or equal to n, and relatively prime to n} All groups of order 4 are Abelian. There are only two such groups: Z 4 and U(4).

NON- ABELIAN GROUPS ØSome examples of Non-Abelian groups are: • Dn, the transformations on a regular n-sided figure under function composition • (GL, n), the non-singular square matrices of order n under matrix multiplication • (SL, n), the square matrices of order n with determinant = 1 under matrix multiplication • Sn, the permutation groups of degree n under function composition • An, the even permutation groups of degree n under function composition

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