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A recent question on the history of Galois theory wasn't the most satisfactory. But the historical issues do seem quite attractive. They relate to innovation, and to exposition. There is a perspective …

Topics in Complex Function Theory, Abelian Functions and Modular Functions of Several Variables by C. L. Siegel is a standard reference using complex function theory. There are older works (e.g. H. F. …

Your argument fails because the bracket can (sometimes) take pairs of elements into the centre. Therefore the direct sum as vector spaces isn't necessarily a direct sum of Lie algebras. For nilpotent …

By looking mod p for some primes you are actually testing a condition more relevant to determining whether the subgroup generated is Zariski-dense in the symplectic group. This is a kind of heuristic …

The Wikipedia article Odd order theorem is worth reading.

Q1: A typical Riemann surface has no holomorphic automorphisms, and this implies a negative answer to Q2. I don't see that Q3 can work: the p-subgroups surely don't uniquely determine the group in gen …

The type of Jacobian (generalised) discussed in Serre's book "Groupes algébriques et corps de classes" provides a large class of examples. They arise, in a sense, from the universality of the Albanese …

The center of a group cannot be of prime index, because a non-central element must fail to commute with something, which therefore cannot be one of its powers.

A Möbius transformation (http://en.wikipedia.org/wiki/Möbius_transformation) is usually considered as acting on the projective line, which is associated to a two-dimensional vector space. Roughly spea …

I don't really see why there should be general results. We can take W to be a vector space over a finite field, and then you are asking about equivalence of linear representations of G on W. The data …

Such questions formed a substantial part of the classical "theory of equations", before Galois theory was formulated. For example, there is a book by Burnside on theory of equations, that is easy to f …

If there were a really simple way we wouldn't need the concept of "gimbal lock" (http://en.wikipedia.org/wiki/Gimbal_lock). In other words the manifold in question is compact but isn't the 3-torus, an …

"Broken symmetry", rather than complete and utter asymmetry, is an important concept for physicists. See http://en.wikipedia.org/wiki/Symmetry_breaking. I suppose this all goes back to Buridan's ass ( …

See http://en.wikipedia.org/wiki/N-group_%28finite_group_theory%29 where it talks about minimal simple groups. You'd need to check in particular that the example of 2x2 projective special linear group …