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We show that the hierarchical model at finite volume has a symmetry groupwhich can be decomposed into rotations and translations as the familiarPoincar\'e groups. Using these symmetries, we show that the intricate sumsappearing in the calculation of the high-temperature expansion of the magneticsusceptibility can be performed, at least up to the fourth order, usingelementary algebraic manipulations which can be implemented with a computer.These symmetries appear more clearly if we use the 2-adic fractions to labelthe sites. We then apply the new algebraic methods to the calculation ofquantities having a random walk interpretation. In particular, we show that theprobability of returning at the starting point after $m$ steps has poles at$D=-2,-4,....-2m$ , where $D$ is a free parameter playing a role similar to thedimensionality in nearest neighbor models.Comment: 24 Pages, includes 2 short Mathematica programs appended after "/end"  uses phyzzx.te

The high-temperature expansion of the hierarchical Ising model: From Poincaré symmetry to an algebraic algorithm