Diagonal subgroup
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In the mathematical discipline of group theory, for a given group G,Script error: No such module "Check for unknown parameters". the diagonal subgroup of the n-fold direct product G nScript error: No such module "Check for unknown parameters". is the subgroup
This subgroup is isomorphic to G.Script error: No such module "Check for unknown parameters".
Properties and applications
- If GScript error: No such module "Check for unknown parameters". acts on a set X,Script error: No such module "Check for unknown parameters". the n-fold diagonal subgroup has a natural action on the Cartesian product X nScript error: No such module "Check for unknown parameters". induced by the action of GScript error: No such module "Check for unknown parameters". on X,Script error: No such module "Check for unknown parameters". defined by
- If GScript error: No such module "Check for unknown parameters". acts nScript error: No such module "Check for unknown parameters".-transitively on X,Script error: No such module "Check for unknown parameters". then the nScript error: No such module "Check for unknown parameters".-fold diagonal subgroup acts transitively on X n.Script error: No such module "Check for unknown parameters". More generally, for an integer k,Script error: No such module "Check for unknown parameters". if GScript error: No such module "Check for unknown parameters". acts knScript error: No such module "Check for unknown parameters".-transitively on X,Script error: No such module "Check for unknown parameters". GScript error: No such module "Check for unknown parameters". acts kScript error: No such module "Check for unknown parameters".-transitively on X n.Script error: No such module "Check for unknown parameters".
- Burnside's lemma can be proved using the action of the twofold diagonal subgroup.
See also
References
- Script error: No such module "citation/CS1"..