Introduction to Group Theory [Lecture Notes] by Jürgen Bierbrauer

By Jürgen Bierbrauer

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MA 462 (version eight Feb 1999)

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De nition. The group PGL(n; p) = GL(n; p)=Z is the projective general linear group. It has a faithful transitive action on the 1-dimensional subspaces. Let us determine the orders of the linear groups. Fix a basis, for example the standard basis e1; e2 ; : : : ; en: A linear function (matrix) is uniquely determined by the images of the ei (the image of ei is row number i of the matrix). This matrix will be invertible if and only if the image of our basis forms a basis again. We conclude that GL(n; p) is in bijection with the ordered bases of our space IFpn: Let us count these bases: The rst vector v1 is an arbitrary nonzero vector.

This means that for every i; j 2 there is some g 2 G such that (g) : i 7! 4. De nition. Let : G ! S be a permutation representation and i 2 : The stabilizer of i is de ned as Gi = fg : g 2 G; (g) : i 7! 5. Proposition. Let : G ! S be a permutation representation and i 2 : The stabilizer Gi is a subgroup of G: This is immediate (see the Problems). The following easy theorem is extremely useful for many applications. 6. Theorem (The orbit lemma). Let : G ! S be a permutation representation, i 2 and O the orbit of i: Then we have G : Gi] = jOj (the length of the orbit is the index of the stabilizer).

S be a permutation representation and i 2 : The stabilizer of i is de ned as Gi = fg : g 2 G; (g) : i 7! 5. Proposition. Let : G ! S be a permutation representation and i 2 : The stabilizer Gi is a subgroup of G: This is immediate (see the Problems). The following easy theorem is extremely useful for many applications. 6. Theorem (The orbit lemma). Let : G ! S be a permutation representation, i 2 and O the orbit of i: Then we have G : Gi] = jOj (the length of the orbit is the index of the stabilizer).

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