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:

1. (ifis finite then).

2. (Ifis finite thenandare coprime).

3. 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.