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One implementation of the algorithm proceeds as follows. Suppose that ${\displaystyle G=\langle X\mid R\rangle }$, where ${\displaystyle X}$ is a set of generators and ${\displaystyle R}$ is a set of relations and denote by ${\displaystyle X'}$ the set of generators ${\displaystyle X}$ and their inverses. Let ${\displaystyle H=\langle h_{1},h_{2},\ldots ,h_{s}\rangle }$ where the ${\displaystyle h_{i}}$ are words of elements of ${\displaystyle X'}$. There are three types of tables that will be used: a coset table, a relation table for each relation in ${\displaystyle R}$, and a subgroup table for each generator ${\displaystyle h_{i}}$ of ${\displaystyle H}$. Information is gradually added to these tables, and once they are filled in, all cosets have been enumerated and the algorithm terminates.

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I'm lost already. Is H a list of words or a list of generators? Does it begin empty, or what? — Tamfang ( talk) 06:13, 2 September 2012 (UTC) reply

H is a subgroup of G, and does not change during the course of the algorithm. H is represented by a list of words h_i from X′ whose images generate the subgroup H. This list and those words do not change during the course of the algorithm. JackSchmidt ( talk)