By Jean-Paul Pier

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However, a definitive result had to wait for the Classification of Finite Simple Groups (CFSG), as we will see in the next section. Using this classification, all multiply transitive groups have been determined. In particular, the only 5transitive groups apart from symmetric and alternating groups are the two Mathieu groups mentioned above. 5 Modern permutation group theory The title of this section is taken from a talk by Michael Aschbacher to the London Mathematical Society in 2001. Aschbacher’s theme was that many questions about finite permutation groups can be reduced to questions about almost simple groups (where a group is said to be almost simple if it is an extension of a non-abelian simple group by a subgroup of its outer automorphism group).

More precisely, the function χ mapping i to the ith column is a representation of M over F. How do the elementary operations affect the matroid representation? We see that row operations on A don’t change M but replace the representation χ by an equivalent representation. ) On the other hand, column permutations and scalar multiplications replace M by an isomorphic matroid; effectively, permutations re-label the elements, while scalar multiplications have no effect at all. So, if we call two matrices A and A CM-equivalent if A is obtained from A by a row operation and a monomial transformation of the columns, we see that CM-equivalence classes of matroids correspond bijectively to both monomial equivalence classes of linear codes, and equivalence classes of representations of matroids, under the natural notions of equivalence in each case.

1 Definition Let G be a permutation group on a set Ω, where |Ω| = n. For each element g ∈ G, we can decompose the permutation g into a product of disjoint cycles; let ci (g) be the number of i-cycles occurring in this decomposition. Now the cycle index of G is the polynomial Z(G) in indeterminates s1 , . . , sn given by Z(G) = 1 c (g) c (g) s11 · · · snn . ∑ |G| g∈G This can be regarded as a multivariate probability generating function for the cycle structure of a random element of G (chosen from the uniform distribution).