# (Dis)similarity between groups and Lie algebras

There are many questions which sound similar or the same for groups and Lie algebras. Some (very few, actually) of those questions have identical solutions and answers. Some have identical answers but very different solutions. Some have different answers and different solutions.

I am not talking about situations where one have a strictly defined (functorial) correspondence of some sort which allows to reduce almost automatically some questions from groups to Lie algebras, like Lie groups-Lie algebras, or Malcev(-like) correspondence.

For simplicity, let us confine to finite groups and finite-dimensional Lie algebras over a field. A few examples of what I have in mind:

1. Suppose a group (resp. Lie algebra) is represented as a (not necessarily direct) product (resp. sum) of two nilpotent subgroups (resp. subalgebras). Is it true that the group (resp. Lie algebra) is solvable?

2. Suppose a group (resp. Lie algebra) is commutative-transitive: $[x,y] = 1$ (resp. 0), $[y,z] = 1$ (resp. 0), and $y \ne 1$ (resp. 0) implies $[x,z] = 1$ (resp. 0). What can be said about its structural properties?

3. (edit: added a couple of hours later) (Lyndon-)Hochschild-Serre spectral sequence connecting group (resp. Lie algebra) (co)homology with (co)homology of its normal subgroup (resp. ideal) and the respective quotient.

As far as I know, both questions 1. and 2. have very similar answers for groups and Lie algebras, in some particular situations admit (almost) identical proofs or fragments of proofs, but in the whole generality the proofs are very different, with group-theoretic proof not utilizing in any way a Lie-algebraic one, and vice versa. As for 3., initially the group- and Lie-algebraic cases were given very similar, but disjoint proofs, and later were interpreted as particular instances of the Grothendieck (?) spectral sequence.

Question. Is it possible, looking on a property of groups/Lie algebras expressed as a formula in the appropriate first- or second-order theory, to predict solely on syntaxical/logical ground, whether this property would convey similar or different results in respective categories? On the other hand, is there some (metamathematical, I dare to say) principle which would make this impossible?

I realize that the question is probably too vague, and I would appreciate any help in making it more precise and interesting.

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My memory is too fuzzy right now for a coherent answer. I will instead give a stream-of-consciousness series of terms which you can use as internet search terms to refine/resolve your question or part of it. Interpretability of first order theories; Zamjatins interpetation of finite graphs into rings to show undecidability of the latter; work of McKenzie, Burris, Jezek on biinterpetability and the lattice of interpretability of theories of certain (equational?) classes of models. Apologies for misremembering. Gerhhard "Universal Algebra Suggests: Sometimes Impossible" Paseman, 2011.06.20 –  Gerhard Paseman Jun 20 '11 at 7:12
Tiny correction: in your definition of commutative-transitivity, you need $y\neq 1$ (and, presumably, $y\neq 0$ respectively). –  HJRW Jun 20 '11 at 10:12
@HW: Yes, of course, thanks. –  Pasha Zusmanovich Jun 20 '11 at 17:14
@Gerhard Paseman: Thanks. Sure, there is a lot of works about interpretability of one theory in another, probably I should look at them closer. –  Pasha Zusmanovich Jun 27 '11 at 8:03

This isn't getting directly at your question, but you might consider looking at Ellis' notion of a multiplicative Lie algebra. This all started in:

G.J. Ellis, On ﬁve well-known commutator identities, J. Aust. Math. Soc. Ser. A 54 (1993), 1–19.

Then just search for "multiplicative Lie algebra" or "multiplicative Lie ring" for more recent papers. Multiplicative Lie rings provide an interesting framework in which to think about some group and Lie algebra results in a unified way.

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Thanks! I was aware of the works of Ellis, though not specifically this one, but somehow it did not occur to me that one may think about them in that direction. Another, probably related in a similar vein thing, is a concept of $\Omega$-groups (which generalizes simultaneously groups and algebras) introduced in 1950s by P.J. Higgins and further developed in 1960s by Kurosh and his students. –  Pasha Zusmanovich Oct 21 '11 at 13:46