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Consider the group G of the sequences of real numbers (the group operation is addition). It contains a subgroup H of bounded sequences.

Is there any nice description of the factor group G/H ?

It is rather uncomfortable to think about this factor group: there is nothing like canonical representative in each coset. Basically, I see no means of thinking about this group except the straightforward definition of the factor group.

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    $\begingroup$ The isomorphism type of this group is all but mysterious: every torsion-free divisible abelian group of uncountable cardinal $\kappa$ is isomorphic to a $\mathbf{Q}$-vector space with basis of cardinal $\kappa$. In particular, the group you're mentioning is isomorphic to $\mathbf{R}$ itself (of course, in a highly non-canonical way). $\endgroup$
    – YCor
    Apr 24, 2023 at 14:34
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    $\begingroup$ There are many similar quotient groups: replace the group of bounded sequences by the group of sequences having at most polynomial growth (of degree at most $k$), at most intermediate growth, etc. Are some of these quotient groups isomorphic? $\endgroup$ Apr 24, 2023 at 14:54

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