Question

Prove that $\text{H=}\left\{\left.\left(\begin{array}{l}\text{a\hspace{0.17em}\hspace{0.17em}b}\\ \text{0\hspace{0.17em}\hspace{0.17em}1}\end{array}\right)\right|\text{a=1\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}}-\text{1\hspace{0.17em},\hspace{0.17em}b∈}\mathbb{Z}\right\}$is a subgroup of $\text{GL}\left(\mathbf{2}\text{\hspace{0.17em},Q}\right)$.

Short answer

Answer

It is proved that $\text{H=}\left\{\left.\left(\begin{array}{l}\text{a\hspace{0.17em}\hspace{0.17em}b}\\ \text{0\hspace{0.17em}\hspace{0.17em}1}\end{array}\right)\right|\text{a=}1\text{\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}}-1\text{\hspace{0.17em},\hspace{0.17em}b∈}\mathbb{Z}\right\}$is a subgroup of $\text{GL}\left(\mathbf{2}\text{\hspace{0.17em},Q}\right)$.

Step-by-step solution

Step by Step Solution Step 1: Required Theorem

A non-empty subset H of group G is a subgroup of G provided that

$$\begin{gathered} \text{(i)Ifa,b}\in \text{H,thenab}\in \text{H;and} \\ \left(ii\right)if\text{a}\in {\text{Hthena}}^{-\mathbf{1}}\in \text{H} \end{gathered}$$

Prove that, H=a  b0  1 a=1  or  −1 , b∈ℤ is a subgroup of GL(2,Q)

Suppose, H is a non-empty subset of GL (2,Q).

$$\begin{gathered} Let\left(\begin{array}{l}{a}_1{b}_1\\ 01\end{array}\right),\left(\begin{array}{l}{a}_2{b}_2\\ 01\end{array}\right)\in H.then, \\ \left(\begin{array}{l}{a}_1{b}_1\\ 01\end{array}\right)\left(\begin{array}{l}{a}_2{b}_2\\ 01\end{array}\right)=\left(\begin{array}{l}{a}_1{a}_2{a}_1{b}_2+b_1\\ 01\end{array}\right)\in H \end{gathered}$$

Therefore, H is closed under group operation.

$$\begin{gathered} Andif\left(\begin{array}{l}ab\\ 01\end{array}\right)\in H,then \\ {\left(\begin{array}{l}ab\\ 01\end{array}\right)}^{-1}=\left(\begin{array}{l}a-b\\ 01\end{array}\right)\in H \end{gathered}$$

Thus, we see that H is closed under inverses.

Hence, as the two conditions listed under theorem 7.11 are satisfied, we can conclude that H is a subgroup of GL(2,Q).