Groups which are not Lie groups

Question

Question: What are examples of set-theoretic groups which do not admit any Lie group structure?

Note that this is different from asking for "a topological group which is not a Lie group", since here we would need to show that it is not a Lie group for any topology.

(Definition: I am assuming a Lie group is a second-countable Hausdorff locally-Euclidean space with a smooth structure for which the product and inversion maps are smooth.)

Note that any group with at most countably many elements can be given the structure of a $0$-dimensional Lie group with the discrete topology. Examples would thus have to be uncountable.

The edited version seems not to be a duplicate now. This question is related. — Dietrich Burde, Sep 22 '16 at 14:29
Kind of a boring example, but according to the convention that manifolds are second countable, locally Euclidean, Hausdorff spaces, it follows that every manifold has cardinality at most that of the real line. So, taking any group with more than continuum-many elements gives an example. — Mike F, Sep 22 '16 at 16:33

Moishe Kohan · Accepted Answer · 2016-09-22T15:10:29.193

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Take an uncountable direct sum of, say, finite cyclic groups.

edited Sep 22 '16 at 15:10

answered Sep 22 '16 at 14:54

Moishe Kohan

97,719

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Why can't this be given the structure of a Lie group? – Najib Idrissi Sep 22 '16 at 14:58
@NajibIdrissi: Because of the known structure of abelian Lie groups. The connected component of the identify is a product of a compact torus and $R^n$, hence, it cannot be torsion. – Moishe Kohan Sep 22 '16 at 15:08

Groups which are not Lie groups

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