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.
(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. The group $\mathbb{Z}_2\times\mathbb{Z}_2$ is abelian, but no _____ ________ ___ _________ _________ _______ ___ ______ ________ _____.
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