What is the subgroup of $GL_{2}(\mathbb{F}_{p})$ that is generated by $ $$SL_{2}(\mathbb{F}_{p})$ and $H_{2}$ where $$ H_{2}=\left\{ \left(\begin{array}{cc} a & b\\ 0 & c\end{array}\right):a,c\in\mathbb{F}_{p}^{\times},b\in\mathbb{F}_{p}\right\} $$
Group generated by two subgroups
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abstract-algebra
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0Remember that the determinant is linear in each of the rows. If you start with a matrix in $GL_2$, can you modify it slightly to get something with determinant $1$? – 2011-08-25