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By Frédérique Oggier

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Proof. The idea of the proof is to actually exhibit a subgroup of G of order pr . For that, we need to define a clever action of G on a carefully chosen set X. Take the set X = {subsets of G of size pr } and for action that G acts on X by left multiplication. This is clearly a welldefined action. We have that |X| = pr m pr which is not divisible by p (by the previous lemma). Recall that the action of G on X induces a partition of X into orbits: X = ⊔B(S) 42 CHAPTER 1. GROUP THEORY where the disjoint union is taken over a set of representatives.

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. Every group acts on itself by conjugation. Let us write this action as g · x = gxg −1 . Let us check the action is actually well defined. First, we have that h · (g · x) = h · (gxg −1 ) = hgxg −1 h−1 = (hg)xg −1 h−1 = (hg) · x. As for the identity, we get 1 · x = 1x1−1 = x. 36 CHAPTER 1. GROUP THEORY Similarly to the notion of kernel for a homomorphism, we can define the kernel of an action.

We can however define an invariant of a permutation, called the parity. 20. An element of Sn is said to be even if it can be expressed as a product of an even number of transpositions. It is said odd otherwise. For this definition to make sense, parity of an element of Sn should be unique, which it is. 24. For n ≥ 2, every element of Sn has a unique parity, even or odd. Proof. To prove this, we need to introduce some ordering on the permutations. We call the switching number of a permutation σ the number of ordered pairs (i, j) with i < j but σ(i) > σ(j).

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Algebraic Methods (November 11, 2011) by Frédérique Oggier

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