Regular Simplices Inscribed into the Cube and Exhibiting a Group Structure

For \( n \in \mathbb{N} \), we interpret the vertex set \( W_n \) of the \( n \)-cube as a vector space over the field \( \mathbb{F}_2 \) and prove that a regular \( n \)-simplex can be inscribed into the \( n \)-cube such that its vertices constitute a subgroup of \( W_n \) if and only if \( n+1 \) is a power of 2. Furthermore, a connection to the theory of Hamming Codes will be established.