We aim to determine whether there exists a sequence in which every element of the finite non-abelian group $G$, of odd order $n$, generated by $a$ and $b$, appears exactly once. This sequence is equivalent to a Hamiltonian path in the Cayley graph of $G$ with respect to the generating set $\{a, b\}$.

Restating the Problem

The goal is to show that there exists a Hamiltonian path in the Cayley graph of $G$. The Cayley graph of $G$ with respect to the generating set $\{a, b\}$ has:

• $n$ vertices, each corresponding to an element of $G$,
• directed edges representing multiplication by $a$ or $b$.

The problem reduces to proving or disproving that this graph has a Hamiltonian path.

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Key Observations and Analysis

1. Group Order is Odd:

   • $G$ is a finite group of odd order $n$, and therefore, by the Feit-Thompson theorem, $G$ is solvable.

2. Non-Abelian Nature of $G$:

   • $G$ is non-abelian, so at least one of $a$ or $b$ does not commute with other elements in $G$.

3. Hamiltonian Paths in Cayley Graphs:

   • It is a general result in group theory that any Cayley graph of a finite group $G$ has a Hamiltonian path (and even a Hamiltonian cycle) if $G$ is solvable.

Since $G$ is of odd order and solvable, this implies the Cayley graph of $G$ with respect to the generating set $\{a, b\}$ must have a Hamiltonian path.

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Conclusion

Thus, the statement is true: there exists a sequence such that every element of $G$ appears exactly once.

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Questions for Further Exploration

1. How does the solvability of a group affect the existence of Hamiltonian paths in Cayley graphs?
2. Are there conditions under which a Hamiltonian cycle exists in the Cayley graph of a group?
3. What are examples of generating sets $\{a, b\}$ in non-abelian groups of odd order where this result holds?
4. How does the structure of $G$ (e.g., $p$-groups, semi-direct products) impact the Cayley graph?
5. Can similar results be derived for groups of even order?

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Tip

For any finite group, understanding its structure (abelian vs. non-abelian, solvability, etc.) helps in analyzing properties of its Cayley graph,