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College Algebra Exam Review 230

College Algebra Exam Review 230 - 240 5 ACTIONS OF GROUPS...

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240 5. ACTIONS OF GROUPS Definition 5.1.15. Consider the action of a group G on its subgroups by conjugation. The stabilizer of a subgroup H is called the normalizer of H in G and denoted N G .H/ . According to Corollary 5.1.14 , if G is finite, then the number of dis- tinct subgroups xHx 1 for x 2 G is OEG W N G .H/Ł D j G j j N G .H/ j : Since (clearly) N G .H/ H , the number of such subgroups is no more than OEG W . Definition 5.1.16. Consider the action of a group G on itself by conjuga- tion. The stabilizer of an element g 2 G is called the centralizer of g in G and denoted Cent .g/ , or when it is necessary to specify the group by Cent G .x/ . Again, according to the corollary the size of the conjugacy class of g , that is, of the orbit of g under conjugacy, is OEG W Cent .g/Ł D j G j j Cent .g/ j : Example 5.1.17. What is the size of each conjugacy class in the symmetric group S 4 ? Recall that two elements of a symmetric group S n are conjugate in S n precisely if they have the same cycle structure (i.e., if when written as a
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