A subset $\left\{ x_{ 1},x_{ 2},\hdots,x_{ d}\right\}$ of a group $G$ \emph{ invariably generates} $G$ if $\left\{ x_{ 1}^{ g_{ 1}},x_{ 2}^{ g_{ 2}},\hdots,x_{ d}^{ g_{ d}}\right\}$ generates $G$ for every $d$-tuple $(g_{ 1},g_{ 2}\hdots,g_{ d})\in G^{ d}$. We prove that a finite completely reducible linear group of dimension $n$ can be invariably generated by $\left\lfloor \frac{ 3n}{ 2}\right\rfloor$ elements. We also prove tighter bounds when the field in question has order $2$ or $3$. Finally, we prove that a transitive [respectively primitive] permutation group of degree $n\geq 2$ [resp. $n\geq 3$] can be invariably generated by $O\left(\frac{ n}{ \sqrt{ \log{ n}}}\right)$ [resp. $O\left(\frac{ \log{ n}}{ \sqrt{ \log{ \log{ n}}}}\right)$] elements.
Categories: math.GR