Question

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

Short answer

It is proved that ${\mathrm{G}}^{*}=\left\{\left(\mathrm{a},\mathrm{e}\right)|\mathrm{a}\in \mathrm{G}\right\}$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 $\left(\mathrm{a},\mathrm{e}\right),\left(\mathrm{b},\mathrm{e}\right)\in \mathrm{G}\times \mathrm{H}$. Then,

role="math" localid="1655881124391" $\begin{array}{rcl}\left(\mathrm{a},\mathrm{e}\right){\left(\mathrm{b},\mathrm{e}\right)}^{-1}& =& \left({\mathrm{ab}}^{-1},\mathrm{e}\right)\\ & \in & {\mathrm{G}}^{*}\end{array}$

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

$\begin{array}{rcl}\left(\mathrm{g},\mathrm{h}\right)\left(\mathrm{a},\mathrm{e}\right){\left(\mathrm{g},\mathrm{h}\right)}^{-1}& =& \left({\mathrm{gag}}^{-1},{\mathrm{hh}}^{-1}\right)\\ & =& \left({\mathrm{gag}}^{-1},\mathrm{e}\right)\\ & \in & {\mathrm{G}}^{*}\end{array}$

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

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