Index and Sectional Category

Let $G$ be a finite group with order $\mathrm{\mid }G\mathrm{\mid }=\mathrm{\ell }$ and $2\le q\le \mathrm{\ell }$. For a free $G$-space $X$, we introduce a notion of $q$-th index of $(X,G)$, denoted by ${\text{ind}}_q(X,G)$. Our concept is relevant in the Borsuk-Ulam theory. We draw general estimates for the $q$-th index in terms of the sectional category of the quotient map $X\to X\mathrm{/}G$, denoted by $\text{secat}(X\to X\mathrm{/}G)$. This property connects a standard problem in Borsuk-Ulam theory to current research trends in sectional category. Under certain hypothesis we observed that $\text{secat}(X\to X\mathrm{/}G)={\text{ind}}_2(X,G)+1$. As an application of our results, we present new results in Borsuk-Ulam theory and sectional category.