Algebraic Methods (November 11, 2011) - download pdf or read online

By Frédérique Oggier

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Then G/G1 is simple and cyclic, hence of prime order. We may repeat this construction on the cyclic subgroup G1 , so by induction, we obtain a composition series G = G0 ⊳ G1 ⊳ G2 ⊳ · · · ⊳ Gm = {1} for G with Gi /Gi+1 of prime order pi for each i. Thus n = = = = |G| |G/G1 ||G1 | |G/G1 ||G1 /G2 | · · · |Gm−1 /Gm ||Gm | p1 p2 · · · pm−1 . The uniqueness of the prime decomposition of n follows from the Jordan-H¨older Theorem applied to G. 58 CHAPTER 1. 11 Solvable and nilpotent groups Let us start by introducing a notion stronger than normality.

Suppose that the claim is not true, that is both the number of Sylow p-subgroups np and the number of Sylow q-subgroups nq are bigger than 1. Let us start this proof by counting the number of elements of order q in G. If a Sylow q-subgroup has order q, it is cyclic and can be generated by any of its elements which is not 1. This gives q − 1 elements of order q per Sylow q-subgroup of G. Conversely, if y has order q, then the cyclic group it generates is a Sylow q-subgroup, and any two distinct Sylow q-subgroups have trivial intersection.

Is } ∩ {j1 , . . , jt } is empty. Such a decomposition of a permutation into product of disjoint cycles is true in general. 23. Every element of Sn can be expressed uniquely as a product of disjoint cycles, up to ordering of the cycles, and notational redundancy within each cycle. Furthermore, every cycle can be written as a product of transpositions. Proof. Let σ be an element of Sn . Choose any index i1 ∈ {1, . . , n}. By applying σ repeatedly on i1 , we construct a sequence of elements of {1, .

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Algebraic Methods (November 11, 2011) by Frédérique Oggier

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