Discussion
Consider the following statements :
(I) If G is a finite group that has only 2 conjugate classes, then |G| = 2.
(II) If G is a group and \(a \in Z(G)\), the center of G, then the conjugacy class of a is {e, a}, where e is the identity of G
Which of the following is true ?
Answer: A
Only (I) is true.
(I) The class equation of a group with 2 classes reads \(|G| = 1 + k\), where k is the size of the other class, and k divides |G|. So \(k \mid k + 1\), forcing \(k = 1\) and \(|G| = 2\).
(II) If a is in the centre, then \(gag^{-1} = a\) for every g, so its conjugacy class is {a} alone, not {e, a}. The identity is in its own class, separate from a unless \(a = e\).
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