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Is the following true regardless of the vector space $V$:$$\text{P}\Gamma\text{L}(V)\cong\text{PGL}(V)\rtimes\langle\sigma\rangle~?$$

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Basically yes, except that what your notation only makes sense if ${\rm dim}(V)=d$. You would normally either just write ${\rm PGL}(V)$ or else ${\rm PGL}(d,K)$ where $K$ is a field. –  Derek Holt Mar 20 '13 at 9:51
Right! Didn't realise overlapping two different notations. Thanks anyway. –  Easy Mar 20 '13 at 12:19

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