I struggle to understand the notion of a standard coset table. In particular, I do not understand what the GAP function 'StandardizeTable' really does. My guess is that it relabels the cosets such that the resulting coset table is standard. My question is: Can it happen that, during this standardization procedure, the coset $1$ (which should correspond to the given subgroup) is relabeled to something else? Or is the coset $1$ always fixed?
2026-03-26 22:13:37.1774563217
Standardization of coset tables in GAP
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Coset 1 is special, as it is the only coset that forms a subgroup. It therefore is fixed, when standardizing. The numbering of the other cosets is somewhat arbitrary (and will depend on how the coset table was obtained). Standardization removes this arbitrariness.
Standardization assumes that the cosets (different from 1) are re-numbered according to the order in which they first arise in the coset table. (This will still depend on the generating set used to represent the group.) That is, the first rows of the coset table will look rather like 2,3,4,5,... rather than randomly chosen.
The main reason for doing this (apart from tidyness) is that it leaves a unique coset table and thus allows for testing equality of subgroups through the associated standardized coset tables.
If you only work with one subgroup (or table) there is basically no need for standardization.