Which of the following statements are true and which are false? Justify your answer with a short proof or a counter example.

(a) Every subgroup of S₃ is normal.

(b) Every abelian group is cyclic.

(c) In a ring with unity, the sum of any two units is a unit.

(d) If a field has characteristic p, p a prime, the field is finite.

(e) If every element in group has finite order, the group is finite.

To find the order of each element in \( U(15) \), we first need to determine the elements in \( U(15) \). 

The set \( U(15) \) consists of all integers less than 15 that are coprime with 15. The numbers that are coprime with 15 are those that have no common divisors with 15 other than 1. The prime factorization of 15 is \( 3 \times 5 \), so we exclude all multiples of 3 and 5 from the list of numbers less than 15. 

Thus, the elements of \( U(15) \) are: 
\[
U(15) = \{1, 2, 4, 7, 8, 11, 13, 14\}
\]

Now, we calculate the order of each element in \( U(15) \). The order _______ ___ ____ ________ _____ _____.
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