Real vector space/Bijective linear mapping/Orientation-preserving/Exercise

Let ${}V$ be a finite-dimensional real vector space, and let ${}G=\operatorname {GL} _{}{\left(V\right)}$ denote the group of all bijective linear mappings on ${}V$. Show that the set of the orientation-preserving mappings is a normal subgroup in ${}G$. What relation exists to the modulus of the determinant?

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