Smallest Possible Non-Trivial Group Containing Elements x and y

To find the smallest possible non-trivial group containing elements x and y, we need to analyze the given conditions and determine the relations between x and y.

Given Conditions:
1. x^7 = y^2 = e (identity element)
2. yx = x^4y

Analysis:
Let's analyze the given conditions step by step to understand the relations between x and y.

Condition 1: x^7 = y^2 = e
This condition tells us that the order of x is 7, and the order of y is 2. In other words, the smallest positive integer n such that x^n = e is 7 for x, and 2 for y.

Condition 2: yx = x^4y
This condition represents the group operation between x and y. It tells us that y followed by x is equal to x raised to the power of 4 followed by y. In other words, yx = x^4y is the defining relation for the group.

Determining the Group:
Now, let's determine the group that satisfies the given conditions. We can start by considering the smallest possible non-trivial group that contains elements x and y.

Case 1: x and y are distinct elements.
If x and y are distinct elements, then the smallest possible non-trivial group containing them is a group of order 14. This group can be represented as ⟨x, y | x^7 = y^2 = e, yx = x^4y⟩.

Case 2: x and y are the same element.
If x and y are the same element, then we need to find the smallest possible non-trivial group that satisfies the given conditions.

Since x^7 = e and y^2 = e, the order of x is 7 and the order of y is 2. Therefore, the smallest possible non-trivial group containing x and y as the same element is a cyclic group of order lcm(7, 2) = 14.

Conclusion:
The order of the smallest possible non-trivial group containing elements x and y such that x^7 = y^2 = e and yx = x^4y is 14. This group can be represented as ⟨x, y | x^7 = y^2 = e, yx = x^4y⟩.