Group theory MOC

Direct product of groups

The (external) direct product is the categorical product in . Given two groups , their product is their Cartesian product with the group operation such that #m/def/group

for any and . It follows that and . This generalized to arbitrarily large products

The projections are split epic.

Internal direct product

Noting ^P3, a related internal construction occurs when there exist normal subgroups such that and . #m/def/group This motivates generalisation to the Semidirect product (both external and internal), where only one group need be normal.

Properties

1. If is the trivial group, P1
2. Clearly .
4. . Usually this is stated as . However it is not generally true that given we have .

#state/tidy | #lang/en | #SemBr