2405. ACTIONS OF GROUPSDefinition 5.1.15.Consider the action of a groupGon its subgroups byconjugation. The stabilizer of a subgroupHis called thenormalizerofHinGand denotedNG.H/.According to Corollary5.1.14, ifGis finite, then the number of dis-tinct subgroupsxHx1forx2GisOEGWNG.H/ŁDjGjjNG.H/j:Since (clearly)NG.H/H, the number of such subgroups is no morethanOEGWHŁ.Definition 5.1.16.Consider the action of a groupGon itself by conjuga-tion. The stabilizer of an elementg2Gis called thecentralizerofginGand denoted Cent.g/, or when it is necessary to specify the group byCentG.x/.Again, according to the corollary the size of the conjugacy class ofg,that is, of the orbit ofgunder conjugacy, isOEGWCent.g/ŁDjGjjCent.g/j:Example 5.1.17.What is the size of each conjugacy class in the symmetricgroupS4?Recall that two elements of a symmetric groupSnare conjugate inSnprecisely if they have the same cycle structure (i.e., if when written as a
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