Prove that the cayley digraph given in example 6 does not

Question

This assignment is a mathematical paper rather than a homework assignment. In particular do no state a problem and jump right into a solution. Segue naturally into the problems. This is quite terse, so more words and explanations are needed and it should flow like a report or paper. Some background on theorems and properties used should be provided for the reader and flow into it the assigned problems one at a time with something like "Consider the Cayley graph of...Let's determine if this has a Hamilton circuit".

When you provide background and theorems, use your own numbering. Do not say, for example "Theorem 14.4". Assume the reader does not own the textbook.

Show that the Cayley digraph given in Example 7 has a Hamiltonian path from e to a.

Example 7: Q₄ = {a,b |a⁴ = e, a² = b², b^-1ab = a³}

Prove that the Cayley digraph given in Example 6 does not have a Hamiltonian circuit. Does it have a Hamiltonian path?

Example 6: A₄ = {(12)(34),(123}