When you write

“Now a group action is faithful if and only if the homomorphism to the group of transformations is an injection, which (as is easy to show and I’m sure you’ve seen) is the case if and only if the kernel of the homomorphism consists of more than just the identity element. ”

I guess it should be: “if and only if the kernel of the homomorphism consists of \emph{no} more than just the identity element”?

*Many thanks — corrected now.*

This is a game I played with students at a revision session on Group Actions at an Open University Pure Mathematics summer school.

]]>And then local models are glued along groups also, for instance the fundamental group of a space glues open subsets of its universal cover and combined with the local group it gives informative and universal (and natural and canonical) ways of going from one point to another.

This idea taken broadly is homotopy theory, in a sense.

Computationally this is very useful, there is a whole industry built around torus actions in symplectic/algebraic geometry (in mechanics).

And homotopy also works in algebraic geometry (with the help of heavy machinery).

Also, results like Hilbert’s 5th problem show that there are deep insights that can be neatly summarized (though many have said that this particular result is not very useful -but this seems to be changing).

]]>A simple general method to do this is to find an action of the group such that you can prove using other means that the two words are not equal. For example, if you can find a group action with respect to which the two words have different numbers of fixed points, then they’re clearly not equal. Similarly if you can find a linear representation with respect to which the two words have different traces or determinants. I use essentially this idea in this math.SE answer to show that the product of two elements of fixed finite orders in a group need not have finite order in general.

(The relationship between a generators-and-relations presentation of a group and its actions is closely analogous to the relationship between syntax and semantics of a logic. There the analogous discussion is this: it is straightforward to show that one statement implies another syntactically, just by writing down a derivation of one from the other, but hard to show that one statement does *not* imply another. A simple general method to do the latter is to find a model in which one statement is true and the other is false.)

This leads to a process of research bit by bit: constructing objects and studying their properties in steps, which can be combined and as a whole provide all possible approaches to a problem.

We form ever larger strategies, like having a family of group actions each extracting some information from a group.

The Langlands program and other large projects come to mind of course.

Other axioms of set theory yield other straightforward methods for constructing group actions. Taking cartesian product yield an action of Sym(X) on XxY, and you may have an action of Sym(X) on Y already. Actions on X pass to the power set P(X) (passing from the notation f(x) to f(A), x an element, A a subset), therefore quotients by equivalence relations also yield actions (taking the partition induced by a relation, a subset of P(X)).

These constructions are functors.

For instance Galois groups act on k, therefore on k^n, then on projective spaces, then on projective varieties, and on their quotients, and further functors like cohomology (which themselves are built in smaller (set-theoretic) steps) yield actions.

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