SARSA

 Torsion Free And Mixed Group:

A group in which every element except the identity element 𝑒 has infinite order is 

known as torsion free (𝑎-periodic or locally infinite). A group having elements both of finite as well as 

infinite order is called a mixed group.

Semigroup And Monoid:

A set with an associative binary operation is called a semigroup. 

A semigroup that has an identity element for the binary operation is called monoid.

Note that every group is both a semigroup and a monoid.

Abelian Group:

A group 𝐺 is abelian if its binary operation is commutative. That is,let (𝐺, ∗) be a 

group. Let , 𝑏 ∈ 𝐺 , then 𝐺 is called an abelian group iff 

𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎

Examples

a. The familiar additive properties of integers and of rationals, real and complex numbers show that

ℤ, ℚ, ℝ and ℂ under addition abelian groups.

b. The set ℤ+ under addition is not a group. There is no identity element for + in ℤ+.

c. The set ℤ+ under multiplication is not a group. There is an identity 1, but no inverse of 3.

d. The familiar multiplicative properties of rational, real and complex numbers show that the sets ℚ+and ℝ+ of positive numbers and the sets ℚ∗,ℝ∗and ℂ∗ of nonzero numbers under multiplication are abelian groups.

e. The set 𝑴𝒎×𝒏ℝ of all 𝑚 × 𝑛 matrices under addition is a group. The 𝑚 × 𝑛 matrix with all entries zero is the identity matrix. This group is abelian.

f. The set 𝑴𝒏ℝ of all 𝑛 × 𝑛 matrices under matrix multiplication is not a group. The 𝑛 × 𝑛 matrix with all entries zero has no inverse.

g. The set of all real-valued functions with domain ℝ under function addition is an abelian group.