Direct factor

Definition in tabular form
A direct factor of a group is defined in the following equivalent ways:

Equivalence of definitions
The equivalence of definitions follows largely from the equivalence of internal and external direct product.

Extreme examples
Every group is the internal direct product of itself and the trivial subgroup. Thus:


 * 1) The trivial subgroup is a direct factor of the whole group.
 * 2) Every group is a direct factor of itself.

High occurrence examples

 * 1) In a finite nilpotent group, all the Sylow subgroups are direct factors. In particular, a finite nilpotent group is the direct product of its Sylow subgroups.
 * 2) In a vector space, any vector subspace is a direct factor, because the complementary subspace can be taken as the complement for an internal direct product.

Relationship with external direct product and restricted external direct product

 * 1) If a group $$G$$ arises as the external direct product of finitely or infinitely many groups $$G_i, i \in I$$, then for any subset $$J \subseteq I$$, the subset of $$G$$ arising as those elements where all coordinates outside of $$J$$ are trivial is a direct factor of $$G$$. The complementary factor can be taken as the subgroup of $$G$$ where all coordinates in $$J$$ are trivial.
 * 2) A similar observation holds for the restricted external direct product.