Word Count : 517

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Let's analyze each statement carefully. ### (a) False If a finite group \( G \) acts on a finite set \( S \), the stabilizer \( G_s \) of an element \( s \in S \) is defined as: \[ G_s = \{ g \in G \mid g \cdot s = s \} \] Different elements in \( S \) can have different stabilizers, so in general, \( G_{s_1} \neq G_{s_2} \) for all \( s_1, s_2 \in S \). However, if \( G \) acts transitively on \( S \), then stabilizers of elements in the same orbit are conjugate to each other, but not necessarily equal. ### (b) True The symmetric group \( S_4 \) consists of all permutations of four elements. The elements of order 3 ___ _____ _________ ______ ___.
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