Question

If $\mathbf{G}$ is a group, show that$<e>$ and$\mathbf{G}$ are normal subgroups.

Short answer

It is proved that and$\mathrm{G}$ is a normal subgroup of$\mathrm{G}$ .

Step-by-step solution

Prove that <e> is normal subgroup

Since it is given that$\mathrm{G}$ be a group. Assume that $\mathrm{g}\in \mathrm{G}$, this implies that:

$\mathrm{g}\left\{\mathrm{e}\right\}=\left\{\mathrm{g}\right\}=\left\{\mathrm{e}\right\}\mathrm{g}$

Therefore, we can say that$\left\{\mathrm{e}\right\}$ is normal group.

Prove that G is normal subgroup

Since it is given that$\mathrm{G}$ be a group. Assume that$\mathrm{g}\in \mathrm{G}$ , this implies that:

$\begin{array}{rcl}\mathrm{gG}& =& \left\{\mathrm{gh}:\mathrm{h}\in \mathrm{G}\right\}\\ & =& \mathrm{G}\\ & =& \left\{\mathrm{hg}:\mathrm{h}\in \mathrm{G}\right\}\\ & =& \mathrm{Gg}\end{array}$

Therefore, we can say that $\mathrm{G}$is normal group.

Hence, it is proved that and $\mathrm{G}$is a normal subgroup of $\mathrm{G}$.