Total finite order elements in (R ) group

A group is a mathematical structure consisting of a set of elements and an operation that combines any two elements to produce a third element. The real numbers (R) form a group under the operation of addition.

Finite order elements

In a group, an element has a finite order if there exists a positive integer n such that raising the element to the power of n gives the identity element of the group. The identity element in the group of real numbers under addition is 0.

To find the total number of finite order elements in the (R) group, we need to consider the different possible orders of elements.

Order 1
The identity element is the only element in the group with order 1. In the case of (R) group, the identity element is 0.

Order 2
To find elements with order 2, we need to find elements that satisfy the equation a + a = 0. Solving this equation, we find that the only element with order 2 is -a, where a is any non-zero real number.

Order 3
To find elements with order 3, we need to find elements that satisfy the equation a + a + a = 0. Solving this equation, we find that there are no elements with order 3 in the (R) group.

Order 4
To find elements with order 4, we need to find elements that satisfy the equation a + a + a + a = 0. Solving this equation, we find that there are no elements with order 4 in the (R) group.

Order 5 and higher
For orders higher than 4, we can similarly find that there are no elements in the (R) group that satisfy the corresponding equations.

Total finite order elements
Therefore, the total number of finite order elements in the (R) group is 1 (order 1) + 1 (order 2) = 2.

Summary
- The (R) group is a group of real numbers under addition.
- Finite order elements are elements that satisfy a particular equation in the group.
- The total number of finite order elements in the (R) group is 2, consisting of the identity element (order 1) and -a (order 2).