Question

Determine, up to isomorphism, all the finite groups with exactly 2 conjugacy classes.
(b) Is there a finite group with class equation 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2?
(c) Compute the following:
equation a) equation equation b) equation

09 Jan 2026
Answer :
Word Count : 563
(a) Let (G) be a finite group with exactly 2 conjugacy classes. One of these classes must be the identity element (e), which forms a class by itself. Let the other class have size (|G|-1). By the class equation, (|G| = 1 + (|G|-1)), which is consistent. Since the size of a conjugacy class of an element (x) is ([G : C_G(x)]), where (C_G(x)) is the centralizer of (x), the class of size (|G|-1) implies ([G : C_G(x)] = |G|-1). This forces (|C_G(x)| = \frac{|G|}{|G|-1}). But (|C_G(x)|) must be an integer, so (|G|-1) divides (|G|). The only positive integer solution is (|G|=2). Thus, up to isomorphism, the only finite group with exactly 2 conjugacy classes is the cyclic group of order 2, (C_2). --- (b) Suppose a finite group has class equation [ 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2 = 12. ] The identity is always a class of size 1. There is another class of size 1, so there exists a nontrivial central element. Let (Z(G)) be the center. Then
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