Which of the following statements are true and which are false? Justif

Question

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.

(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.

14 Feb 2025

Answer :

Word Count : 743

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 of an element \( a \) in \( U(15) \) is the smallest integer \( k \) such that \( a^k \equiv 1 \pmod{15} \).

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