Which of the following statements are true and which are false? Give r

Question

Which of the following statements are true and which are false? Give reasons for your answer.
(a) If a finite group G acts on a finite set S, then $G_{s_1} = G_{s_2}$ for all $s_1, s_2 \in X$ .
(b) There are exactly 8 elements of order 3 in S₄.
(c) If $F = \mathbb{Q}(\sqrt[5]{2}, \sqrt[3]{5})$ , then $[F : \mathbb{Q}] = 8$ .
(d) $\mathbb{F}_7(\sqrt{3}) = \mathbb{F}_7(\sqrt{5})$ .
(e) For any $\alpha\in\mathbb{F}_{25}^{<em>}$ , $\alpha \neq 1, \mathbb{F}_{25} = \mathbb{F}_5[\alpha]$ .

09 Jan 2026

Answer :

Word Count : 185

(a) True. By the Orbit-Stabilizer Theorem, if a finite group (G) acts on a finite set (S), then (|G| = |G_s| \cdot |O_s|) for any (s \in S), where (G_s) ______ __________ ___ __________ ______ ______.
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