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In part 1, we review the structure theory of $\mathbb{F} S_n$, the groupalgebra of the symmetric group $S_n$ over a field of characteristic 0. Wedefine the images $\psi(E^\lambda_{ij})$ of the matrix units $E^\lambda_{ij}$($1 \le i, j \le d_\lambda$), where $d_\lambda$ is the number of standardtableaux of shape $\lambda$, and obtain an explicit construction of Young'sisomorphism $\psi\colon \bigoplus_\lambda M_{d_\lambda}(\mathbb{F}) \to\mathbb{F} S_n$. We then present Clifton's algorithm for the construction ofthe representation matrices $R^\lambda(p) \in M_{d_\lambda}(\mathbb{F})$ forall $p \in S_n$, and obtain the reverse isomorphism $\phi\colon \mathbb{F} S_n\to \bigoplus_\lambda M_{d_\lambda}(\mathbb{F})$.  In part 2, we apply the structure theory of $\mathbb{F} S_n$ to the study ofmultilinear polynomial identities of degree $n \le 7$ for the algebra$\mathbb{O}$ of octonions over a field of characteristic 0. We compare ourresults with earlier work of Racine, Hentzel & Peresi, and Shestakov &Zhukavets on the identities of degree $n \le 6$. We use computational linearalgebra to verify that every identity in degree 7 is a consequence of the knownidentities of lower degrees: there are no new identities in degree 7. Weconjecture that the known identities of degree $\le 6$ generate all octonionidentities in characteristic 0.

Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions