Question

Prove that a group is indecomposable if and only if whenever and are normal subgroups such that $\mathit{G}\mathbf{=}\mathit{H}\mathbf{}\mathit{x}\mathbf{}\mathit{K}$, then data-custom-editor="chemistry" $\mathbf{H}\mathbf{=}<e>$or $\mathit{K}\mathbf{=}<e>$

Short answer

It has been proved that, a group is indecomposable if and only if whenever and are normal subgroups such that $G=HxK$, then or .

Step-by-step solution

Definition of Indecomposable

A group $\mathrm{G}$is said to be indecomposable if it is not the direct product of its two proper normal subgroups.

Step 2: Prove H or K equals <e>

Suppose that is indecomposable and $H$and $K$are its normal subgroups such that, .$G=HxK$

Since $G$is indecomposable, it is not the direct product of its two proper normal subgroups.

But $G=HxK$, therefore, one of them must be improper subgroup.

Hence, either $H$or K equals.

Prove G is Indecomposable

Suppose $H$that $K$and are normal subgroups $G$of such that $G=HxK$, and either or .

It is clear from the supposition that,-$G$is not the direct product of its two proper normal subgroups.

Hence, $G$is indecomposable.

Thus, it can be concluded that a group $G$is indecomposable if and only if whenever $H$and $K$are normal subgroups such that $\mathrm{G}=\mathrm{H}\mathrm{x}\mathrm{K}$, then,or.