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Generalised Cogrowth series, random walks, and the group determinant


We associate to a group G a series that generalises the cogrowth series of G and is related to a random walk on G. We show that the series is rational if and only if G is finite, generalizing a result of Kouksov [Kou]. We show that when G is finite, the series determines G. There are naturally occurring ideals and varieties that are acted on by Aut(G). We study these and generalize this to the context of S-rings over finite groups. There is an associated representation of Aut(G) and we characterize when this is irreducible.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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