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.

29 Apr 2025

Answer :

Word Count : 151

(a) False. In $S_3$, the subgroups $\langle(12)\rangle,\ \langle(13)\rangle,\ \langle(23)\rangle$ (order $2$) are not normal because, for example, $(123)(12)(132)=(23)\notin\langle(12)\rangle$. Only $\{e\}$, $A_3=\langle(123)\rangle$ (index $2$), and $S_3$ itself are normal. (b) False. _____ ___ _______ ______ ___ ______ ______ __________ __________ ______ _________.
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