Existence of Group G with 5 Elements of Order 2

Introduction:
In this response, we will explore the existence of a group G that has 5 elements of order 2. We will provide a detailed explanation and reasoning for our answer.

Explanation:
To determine if such a group exists, we need to consider the properties and structure of groups. In a group, the order of an element is defined as the smallest positive integer n such that a^n = e, where a is an element of the group and e is the identity element.

Properties of Elements of Order 2:
- An element of order 2 must be its own inverse, i.e., a^2 = e.
- The identity element e always has order 1.
- If an element a has order 2, then a^-1 = a.

Possible Group Structures:
To find a group with 5 elements of order 2, we can consider different possibilities for the group structure. Let's analyze each case:

Case 1: All Elements Have Order 2:
If all elements in a group have order 2, then the group must be the Klein four-group or V4. However, the Klein four-group has only 4 elements, contradicting the requirement of having 5 elements.

Case 2: Some Elements Have Order 2:
In this case, we consider a group where only some of the elements have order 2. Let's assume there are n elements of order 2 in the group.

Number of Elements:
Since the identity element always has order 1, we have (n+1) elements in the group.

Order of Group:
The order of a group is equal to the sum of the orders of its elements. Therefore, the order of the group is (1 * 1) + (2 * n) = 2n + 1.

Conclusion:
Based on the analysis, we can conclude that there does not exist a group G with 5 elements of order 2. This is because the order of a group must be a positive integer, and 2n + 1 is an odd number. Therefore, it is impossible to have a group with 5 elements of order 2.

Note:
It is important to note that this conclusion is based on the assumption that we are considering finite groups. In the case of infinite groups, it is possible to have a group with 5 elements of order 2, such as the group of integers under addition. However, since the question does not specify whether we are considering finite or infinite groups, we have assumed finite groups in our analysis.