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Problem 8 (Automorphisms Of A Directed Square)
Consider the directed graph $\mathcal{G} = (V,A)$ shown below with vertex set $V = \{1,2,3,4\}$ and arrows (directed edges) $$A = \{(1,2),(2,3),(3,4),(4,1)\}.$$
- How would you define an automorphism of a directed graph?
- List all the automorphisms of this directed graph. You should have 4.
- Is there is an automorphism $a\in \aut(\mathcal{G})$ such that $a^2=a\circ a$ (the composition) is the identity automorphism, but $a$ is not the identity?
- Is there is an $a\in \aut(\mathcal{G})$ such that $a^3=a\circ a\circ a$ is the identity but $a$ is not the identity?
- Is there is an $a\in \aut(\mathcal{G})$ such that every other $b\in \aut(\mathcal{G})$ is of the form $b=a^n$ for some $n$?
- For every $a,b\in \aut(\mathcal{G})$, do we have $a\circ b = b\circ a$?
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