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 ?, then for all .

(b) There are exactly 8 elements of order 3 in  .

(c) If , then .

(d) .

(e) For any 

Let's analyze each statement:

(a) If a finite group \( G \) acts on a finite set \( S \), then \( G_{s_1} = G_{s_2} \) for all \( s_1 , s_2 \in S \).

This statement is false. Here \( G_{s} \) represents the stabilizer of the element \( s \) under the group action. It's not necessarily true that the stabilizers of different elements in the set \( S \) are equal. For instance, consider the group \( G = \{e, (12)\} \) acting on the set \( S = \{1, 2, 3\} \) by permutation. The stabilizers \( G_1 \) and \( G_2 \) are different since \( G_1 = G \) while \( G_2 = \{e\} \).

(b) There are exactly 8 elements of order 3 in \( S_4 \).

This statement is false. In \( S_4 \), there are indeed elements of order 3, ______ ____ ____ ______ ______ ____.
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