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class equation (Theorem)

The conjugacy classes of a group form a partition of its elements. In a finite group, this means that the order of the group is the sum of the number of elements of the distinct conjugacy classes. For an element $ g$ of group $ G$, we denote the centralizer in $ G$ of $ g$ by $ C_G(g)$. The number of elements in the conjugacy class of $ g$ is $ [G:C_G(g)]$, the index of $ C_G(g)$ in $ G$. For an element $ g$ of the center $ Z(G)$ of $ G$, the conjugacy class of $ g$ consists of the singleton $ \{g\}$. Putting this together gives us the class equation

$\displaystyle \vert G\vert = \vert Z(G)\vert + \sum_{i=1}^m [G:C_G(x_i)] $
where the $ x_i$ are elements of the distinct conjugacy classes contained in $ G\setminus Z(G)$.



"class equation" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: conjugacy class

Other names:  conjugacy class formula
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Cross-references: contained, singleton, center, index, centralizer, number, sum, order, finite group, partition, group, conjugacy classes
There are 3 references to this entry.

This is version 6 of class equation, born on 2002-11-27, modified 2007-08-13.
Object id is 3624, canonical name is ConjugacyClassFormula.
Accessed 3922 times total.

Classification:
AMS MSC: 20E45 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Conjugacy classes)
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