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College Algebra Exam Review 240

College Algebra Exam Review 240 - j G j j Cent.g j where...

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250 5. ACTIONS OF GROUPS (b) Another way is to recall that Z 2 Z 2 can be described as a group with four elements e; a; b; c , with each nonidentity element of order 2 and the product of any two nonidentity elements equal to the third. Show that any permutation of f a; b; c g determines an automorphism and, conversely, any automorphism is given by a permutation of f a; b; c g . 5.3.8. Describe the automorphism group of Z n Z n . (The description need not be quite as explicit as that of Aut . Z 2 Z 2 / .) Can you describe the automorphism group of . Z n / k ? 5.4. Group Actions and Group Structure In this section, we consider some applications of the idea of group actions to the study of the structure of groups. Consider the action of a group G on itself by conjugation. Recall that the stabilizer of an element is called its centralizer and the orbit of an element is called its conjugacy class . The set of elements z whose conjugacy class consists of z alone is precisely the center of the group. If G is finite, the decomposition of G into disjoint conjugacy classes gives the equation j G j D j Z.G/ j C X
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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....
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