Isomorphism with GLn(R)

  1. Hello.
    My book offers this statement with no proof, i have been searching in other books with no luck !
    I'm beginning to question whether or not the statement is valid at all !
    Here it goes:
    "Every group G of order n is isomorphic to a subgroup of GLn(R)"
    Could someone please help me out with this?
    I'd greatly appreciate it.
  2. jcsd
  3. morphism

    morphism 2,020
    Science Advisor
    Homework Helper

    Try showing that S_n, the symmetric group on n letters, is isomorphic to a subgroup of GL_n(R). By Cayley's theorem, this will be enough. And like the proof of Cayley's theorem, try to use group actions to prove the assertion.
  4. Alright, that was a brilliant way of approaching it!
    Thank you for clearing my path a bit!
    However I'm afraid that i am not familiar at all with group actions! As for the Cayley theorem proof, i built the isomorphism between two groups using F(x) = ax where a is an element of my random group g and then took it from here.
    Could anyone explain the concept of group actions a bit more?How does it relate to my question?
  5. morphism

    morphism 2,020
    Science Advisor
    Homework Helper

    Group actions are nifty. I highly recommend you look into them.

    For example, here's how one can prove Cayley's theorem using them. Suppose you have a finite group G of n elements. Let G act on itself by left translation, i.e. g(h) -> gh. This induces a homomorphism from G into S_n, whose kernel is {g in G : h=g(h)=gh for all h in G} = {e}, and thus G =~ G/{e} <= S_n. [Note that the homomorphism induced by the action is precisely the F you defined!]

    Now here's how we could approach your problem. Let G be a group of n elements, now considered as a subgroup of S_n. Let {v_1, ..., v_n} be a fixed basis for R^n. Each element of G acts on this basis by permuting the indices. But there is also a corresponding matrix in GLn(R) that "does the same thing", namely the one whose rows are the basis vectors permuted accordingly. This gives us something we can use to get an imbedding of S_n into GLn(R).

    Here's some reading:
  6. Chris Hillman

    Chris Hillman 2,332
    Science Advisor

    Ditto morphism. For a really nice introduction try Neumann, Stoy, and Thompson, Groups and Geometry, Oxford University Press, 1994.
  7. That is totally awesome.
    They actually had a copy of the book at the library!
    The problem now became very easy.
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