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class equation theorem (Theorem)

Let $ G$ be a group acting on a finite set $ X$. Define the set of invariants in $ X$ by the action of $ G$ as $ G_{X}=\{x \in X \quad \lvert \quad gx=x \quad \forall g \in G\}$. Then there are $ H_{1}, ..., H_{r}$ subgroups of $ G$ with $ H_{i} \neq G \quad \forall i$ such that $ \char93 X = \char93 G_{X} + \sum_{i=1}^{r}[G:H_{i}]$



"class equation theorem" is owned by gumau.
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See Also: centralizer

Other names:  class equation

Attachments:
proof of class equation theorem (Proof) by gumau
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Cross-references: subgroups, action, invariants, finite set, group
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This is version 2 of class equation theorem, born on 2004-05-01, modified 2004-05-24.
Object id is 5821, canonical name is ClassEquationTheorem.
Accessed 3607 times total.

Classification:
AMS MSC20D20 (Group theory and generalizations :: Abstract finite groups :: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure)

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