Question

Let $\mathbf{G}$and$\mathbf{H}$ and be groups. Prove that${\mathbf{G}}^{\mathbf{*}}\mathbf{=}\mathbf{\left\{}\mathbf{\right(}\mathbf{a}\mathbf{,}\mathbf{e}\mathbf{)}\mathbf{\mid }\mathbf{a}\mathbf{\in }\mathbf{G}\mathbf{\}}$ is a normal subgroup of$\mathbf{G}\mathbf{\times }\mathbf{H}$ .

Short answer

It is proved that${\text{G}}^{*}=\left\{\right(\mathrm{a},\mathrm{e})\mid \mathrm{a}\in \mathrm{G}\}$is a normal subgroup of$\mathrm{G}\times \mathrm{H}$.

Step-by-step solution

Prove that G* is a subgroup of G×H

It is given that $\mathrm{G}$and $\mathrm{H}$be two groups. We have to show that ${\mathrm{G}}^{*}$is the subgroup of $\mathrm{G}\times \mathrm{H}$.

Assume two elements as $(\mathrm{a},\mathrm{e}),(\mathrm{b},\mathrm{e})\in \mathrm{G}\times \mathrm{H}$. Then,

localid="1657294155350" $$\begin{gathered} (\mathrm{a},\mathrm{e})(\mathrm{b},\mathrm{e}{)}^{-1}=\left({\mathrm{ab}}^{-1},\mathrm{e}\right) \\ \in {\mathrm{G}}^{*} \end{gathered}$$

This implies that${\mathrm{G}}^{*}$is the subgroup of$\mathrm{G}\times \mathrm{H}$.

Prove that G* is a normal subgroup of G×H

Consider the element$(\mathrm{g},\mathrm{h})\in \mathrm{G}\times \mathrm{H}$and $(\mathrm{a},\mathrm{e})\in {\mathrm{G}}^{*}$, this implies that:

role="math" localid="1657294406179" $$\begin{gathered} (\mathrm{g},\mathrm{h})(\mathrm{a},\mathrm{e})(\mathrm{g},\mathrm{h}{)}^{-1}=\left({\mathrm{gag}}^{-1},{\mathrm{hh}}^{-1}\right) \\ =\left({\mathrm{gag}}^{-1},\mathrm{e}\right) \\ \in {\mathrm{G}}^{*} \end{gathered}$$

This implies that ${\mathrm{G}}^{*}$is the normal subgroup of $\mathrm{G}\times \mathrm{H}$.

Hence, it is proved that${\mathrm{G}}^{*}=\left\{\right(\mathrm{a},\mathrm{e})\mid \mathrm{a}\in \mathrm{G}\}$is a normal subgroup of$\mathrm{G}\times \mathrm{H}$.