On stallings' unique factorisation groups

Publisher： Cambridge University Press

E-ISSN： 1755-1633|73|1|27-36

ISSN： 0004-9727

Source： Bulletin of the Australian Mathematical Society, Vol.73, Iss.1, 2006-02, pp. : 27-36

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Abstract

Let Γ be a group and Σ a symmetric generating set for Γ. In 1966, Stallings called Γ a unique factorisation group if each group element may be written in a unique way as a product a1…am, where ai Σ for each i and aiai+1 Σ {1} for each i < m. In this paper we give a complete combinatorial proof of a theorem, not explicitly stated by Stallings in 1966, characterising all such pairs (Γ, Σ). We also characterise the unique factorisation pairs by a certain tree-like property of their Cayley graphs.

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