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
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.
1 is satisfied since 6=2*3.
2 is satisfied since
is abelian so all subgroups are normal and 3 is satisfied.