Symmetric group of countable degree

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

The symmetric group of countable degree is defined as the symmetric group on a set of countable size. This group is denoted $\operatorname {Sym} (\Omega )$ or $S_{\omega }$.

The typical countable set used is $\{1,2,3,\dots \}$, the set of natural numbers.

Group properties

Function	Value	Explanation
abelian group	No
centerless group	Yes	symmetric groups are centerless
nilpotent group	No
solvable group	No
finite group	No
finitely generated group	No
complete group	Yes	symmetric groups on infinite sets are complete
ambivalent group	Yes
rational group	Yes