New PDF release: Algebraic Methods (November 11, 2011)

By Frédérique Oggier

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Extra info for Algebraic Methods (November 11, 2011)

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22. The group G acts on the set X if for all g ∈ G, there is a map G × X → X, (g, x) → g · x such that 1. h · (g · x) = (hg) · x for all g, h ∈ G, for all x ∈ X. 2. 1 · x = x for all x ∈ X. The first condition says that we have two laws, the group law between elements of the group, and the action of the group on the set, which are compatible. 27. Let us consider two examples where a group G acts on itself. 1. Every group acts on itself by left multiplication. This is called the regular action. 2.

Then xgx−1 ∈ K since by assumption K is a normal subgroup of G, and xgx−1 ∈ Gi+1 since Gi+1 ⊳ Gi . Thus xgx−1 ∈ K ∩ Gi+1 which proves that K ∩ Gi+1 ⊳ K ∩ Gi . 10. THE JORDAN-HOLDER THEOREM 53 We now look at the quotient group (K ∩ Gi )/(K ∩ Gi+1 ). Since Gi /Gi+1 is simple, Gi+1 is a maximal normal subgroup of Gi , and thus the only normal subgroups of Gi that contain Gi+1 are Gi and Gi+1 . Recall that K ∩Gi is normal in Gi (it is the kernel of the canonical projection of G to G/K restricted to Gi ), so that we get Gi+1 ⊳ (K ∩ Gi )Gi+1 ⊳ Gi .

I), . . , τ (j), . . , σ(n)) (where the first vector is ordered, but not the second and the third). To understand the effect of the transposition τ on the switching number of σ (that is we are computing the switching number of τ σ and see how it differs from that of σ), we need to remember that we are looking at all the ordered pairs (k, l), k < l, in (1, 2, . . 7. PERMUTATIONS AND GROUP ACTION 1. , i < j), the switching number thus does not change, however when applying τ , the ordering is reversed, and thus t increases by 1, or (b) the ordering is changed, but τ changes again the ordering, so that t decreases by 1.

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Algebraic Methods (November 11, 2011) by Frédérique Oggier


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