Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 7 / 4 / iii

Relabel each duad of each synthematic total by the chosen transposition and compare the resulting synthematic total with the six labelled synthematic totals. For example, $(12)$ interchanges $T_1,T_2$, interchanges $T_3,T_4$, and interchanges $T_5,T_6$. Applying the same calculation to the other adjacent transpositions givesThese formulae use the PDF's numbering of the synthematic totals, which is essential for the particular fixed elements below. Compose permutations from left to right, consistently with the right action on synthematic totals. Multiplying the second and fourth images gives $(23)(45)$, while multiplying the third and fifth gives $(34)(56)$. Hence the two involutionsare fixed by the outer automorphism $\alpha$. Since $\alpha$ is a group automorphism, their product is fixed as well. Applying $u$ and then $v$ yieldsUnder right-to-left composition the product is the inverse cycle, which is also fixed; the displayed convention removes that ambiguity.