Understanding the problem:
We are given that in a non-abelian group, the element 'a' has order 108. We need to find the order of 'a42'.

Key Points:
- The order of an element 'a' in a group is the smallest positive integer 'n' such that a^n = e, where 'e' is the identity element.
- The order of an element 'a' divides the order of the group.
- The order of a product of elements is the least common multiple (LCM) of their individual orders, provided the elements commute.

Solution:

To find the order of 'a42', we need to find the smallest positive integer 'm' such that (a42)^m = e.

Step 1: Simplify (a42)^m
Since 'a' has order 108, we can rewrite (a42)^m as a^(42m).

Step 2: Find the order of a^(42m)
Let's assume the order of a^(42m) is 'k'. Then we have (a^(42m))^k = e.

Step 3: Simplify (a^(42m))^k
Using the property of exponents, we can rewrite (a^(42m))^k as a^(42mk).

Step 4: Equate a^(42mk) to e
Since the order of 'a' is 108, we know that a^108 = e. Equating it to a^(42mk), we have a^(42mk) = e.

Step 5: Find the smallest positive integer 'k'
Since 'a' has order 108, we can rewrite the equation a^(42mk) = e as a^(3mk) = e.
Therefore, the smallest positive integer 'k' that satisfies this equation would be the order of a^(3mk).

Step 6: Find the order of a^(3mk)
Using the property of exponents, we can rewrite a^(3mk) as (a^3)^mk.

Step 7: Find the order of a^3
Since the order of 'a' is 108, we know that a^108 = e. Therefore, the order of a^3 would be 108 divided by the greatest common divisor (GCD) of 3 and 108.

Step 8: Calculate the GCD of 3 and 108
The GCD of 3 and 108 is 3.

Step 9: Calculate the order of a^3
The order of a^3 would be 108 divided by the GCD, which is 108/3 = 36.

Step 10: Calculate the order of a^(3mk)
Since the order of a^3 is 36, the order of a^(3mk) would be the smallest positive integer 'k' such that (3mk) divides 36.

Step 11: Determine the possible values of 'k'
The possible values of 'k' are the divisors of 36, which are 1, 2, 3, 4, 6, 9,