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A major problem in the study of combinatorial aspects of permutation groupsis to determine the distances in the symmetric group $\Sym_n$ with respect to agenerator set. One well-known such a case is when the generator set $S_n$consists of block transpositions. It should be noted that "the blocktransposition distance of a permutation" is the distance of the permutationfrom the identity permutation in the Cayley graph $\Cay$, and "sorting apermutation by block transpositions" is equivalent to finding shortest paths in$\Cay$. The original results in our thesis concern the lower and upper boundson the block transpositions diameter of $\Sym_n$ with respect to $S_n$ and theautomorphism group $\Aut(\Cay)$. A significant contribution is to show how fromthe toric equivalence can be obtained bijective maps on $\Sym_n$ that we call\emph{toric maps}. Using the properties of the toric maps, we discuss the roleof the invariance principle of the block transposition distance within toricclasses in the proof of the Eriksson bound. Furthermore, we prove that$\Aut(\Cay)$ is the product of the right translation group by$\textsf{N}\rtimes\textsf{D}_{n+1}$, where $\textsf{N}$ is the subgroup fixing$S_n$ elementwise, and $\textsf{D}_{n+1}$ is a dihedral group whose maximalcyclic subgroup is generated by the toric maps. Computer aided computationsupports our conjecture that $\textsf{N}$ is trivial. Also, we prove that thesubgraph $\Gamma$ with vertex-set $S_n$ is a $2(n-2)$-regular graph whoseautomorphism group is $\textsf{D}_{n+1}$. We show some aspects of $\Cay$,notably $\Gamma$ has as many as $n+1$ maximal cliques of size $2$, its subgraph$\Gamma(V)$ whose vertices are those in these cliques is a $3$-regularHamiltonian graph, and $\textsf{D}_{n+1}$ acts faithfully on $V$ as a vertexregular automorphism group.

Combinatorial properties of block transpositions on Symmetric groups