Unformatted text preview: j G j j Cent .g/ j ; where Z.G/ denotes the center of G , Cent .g/ the centralizer of g , and the sum is over representatives of distinct conjugacy classes in G n Z.G/ . This is called the class equation . Example 5.4.1. Let’s compute the right side of the class equation for the group S 4 . We saw in Example 5.1.17 that S 4 has only one element in its center, namely, the identity. Its nonsingleton conjugacy classes are of sizes 6, 3, 8, and 6. This gives 24 D 1 C 6 C 3 C 8 C 6 . Consider a group of order p n , where p is a prime number and n a positive integer. Every subgroup has order a power of p by Lagrange’s theorem, so for g 2 G n Z.G/ , the size of the conjugacy class of g , namely, j G j j Cent .g/ j ; is a positive power of p . Since p divides j G j and j Z.G/ j ² 1 , it follows that p divides j Z.G/ j . We have proved the following: Proposition 5.4.2. If j G j is a power of a prime number, then the center of G contains nonidentity elements....
View
Full Document
 Fall '08
 EVERAGE
 Algebra, Group Theory, Prime number, Symmetric group, Group isomorphism, Conjugacy class

Click to edit the document details