Final answer:
To determine whether U₈ and U₁₀ are isomorphic, we review their elements and operations. U₈ lacks an element of order 4 that U₁₀ contains, demonstrating that these groups cannot be structurally identical and hence, are not isomorphic.
Step-by-step explanation:
In order to prove that U₈ is not isomorphic to U₁₀, we can look at the properties of these two groups. U₈, or the group of units modulo 8, consists of the numbers less than 8 that are also relatively prime to 8, which are {1, 3, 5, 7}. U₁₀, or the group of units modulo 10, consists of the numbers that are less than 10 and relatively prime to 10, which are {1, 3, 7, 9}. An isomorphism between two groups exists if there's a bijection between the two groups that also preserves the group operation (in this case, multiplication modulo 8 and 10 respectively).
However, we can see that U₈ does not have an element of order 4, which means there's no element that multiplied by itself four times will result in the identity element of the group. On the other hand, U₁₀ has an element of order 4, which is 3 (since 34 mod 10 = 1). This difference in properties shows that the two groups are not structurally the same, and therefore cannot be isomorphic.