The Internal Direct Product Theorem For Finite Groups
If a group is finite we can modify The Internal Direct Product Theorem as below.
The Internal Direct Product Theorem
Ifandare subgroups of a groupthenis an isomorphism if and only if the following conditions are satisfied:

(ifis finite then).

(Ifis finite thenandare coprime).

andare normal subgroups of
Proof:
so we must prove that all theare distinct. Suppose not, so thatbut thenThe left hand side is inand the right hand side is inso both are equal to e since the intersection is trivial soand All theare distinct therefore andThis implies thatis one to one. Sinceis also onto.
Ifare subgroups of G,is a subgroup of bothandBy Lagrange's Theorem,must divide bothandbut these are coprime sohence
All the conditions of the internal direct product theorem are met.
Example:and
1 is satisfied since 6=2*3.
2 is satisfied since
is abelian so all subgroups are normal and 3 is satisfied.