Question

Let G be the group ${\mathit{S}}_{\mathbf{3}}\mathbf{\times }{\mathrm{\mathbb{Z}}}_{\mathbf{4}}$and let $\mathit{a}\mathbf{=}\left(\left(123\right),2\right)$and $\mathit{b}\mathbf{=}\left(\left(12\right),1\right)$.

Verify that the set $\mathit{T}\mathbf{=}\left\{e=a^0,a^1,a^2,a^3,a^4,a^5,b,ab,a^{2}b,a^{3}b,a^{4}b,a^{5}b\right\}$ consists of 12 distinct elements.

Short answer

It is proved that, T consists of 12 distinct elements.

Step-by-step solution

Given information

It is given that $G=S_3\times {\mathrm{\mathbb{Z}}}_4$.

Let $a=\left(\left(123\right),2\right)$and $b=\left(\left(12\right),1\right)$.

Prove that T consists of 12 distinct elements

The 12 elements can be written explicitly.

And is a subgroup of 6 elements and , since the second component is odd.

Therefore, this contains 12 distinct elements.