Finitary 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 finitary symmetric group of countable degree is defined as the finitary symmetric group on a set of countable size. In other words, it is the subgroup of the symmetric group of countable degree comprising the finitary permutations: the permutations that move only finitely many elements.

Arithmetic functions

Function	Value	Explanation
order	infinite (countable)
exponent	infinite
minimum size of generating set	countable
subgroup rank	countable

Group properties

Property	Satisfied	Explanation
finite group	No
abelian group	No
nilpotent group	No
solvable group	No
locally finite group	Yes
finitely generated group	No
periodic group	Yes
centerless group	Yes
ambivalent group	Yes
rational group	Yes
monolithic group	Yes
one-headed group	Yes
almost simple group	Yes