66 Chapter 1. Groups I What is the stabilizer Gv0 of v0? Aside from the identity, only one g ∈ D8 fixes v0, namely, g = (1 3) hence Gv0 is a subgroup of order 2. (This example can be generalized to the dihedral group D2n acting on a regular n-gon.) Example 1.103. When a group G acts on itself by conjugation, then the orbit O(x) is {y ∈ G : y = axa−1 for some a ∈ G}. Thus, O(x) is called the conjugacy class of x, and it is commonly denoted by xG (we have already mentioned conjugacy classes in Exercise 1.49 on page 44). For example, Theorem 1.9 shows that if α ∈ Sn, then the conjugacy class of α consists of all the permutations in Sn having the same cycle structure as α. As a second example, an element z lies in the center Z(G) if and only if zG = {z} that is, no other elements in G are conjugate to z. If x ∈ G, then the stabilizer Gx of x is CG(x) = {g ∈ G : gxg−1 = x}. This subgroup of G, consisting of all g ∈ G that commute with x, is called the centralizer of x in G. Example 1.104. Every group G acts on the set X of all its subgroups, by conju- gation: if a ∈ G, then a acts by H → aHa−1, where H ⊆ G. If H is a subgroup of a group G, then a conjugate of H is a subgroup of G of the form aHa−1 = {aha−1 : h ∈ H}, where a ∈ G. Since conjugation h → aha−1 is an injection H → G with image aHa−1, it follows that conjugate subgroups of G are isomorphic. For example, in S3, all cyclic subgroups of order 2 are conjugate (for their generators are conjugate). The orbit of a subgroup H consists of all its conjugates notice that H is the only element in its orbit if and only if H G that is, aHa−1 = H for all a ∈ G. The stabilizer of H is NG(H) = {g ∈ G : gHg−1 = H}. This subgroup of G is called the normalizer of H in G. Of course, H NG(H) indeed, the normalizer is the largest subgroup of G in which H is normal. We have already defined the centralizer of an element we now define the cen- tralizer of a subgroup. Definition. If H is a subgroup of a group G, then the centralizer of H in G is CG(H) = {g ∈ G : gh = hg for all h ∈ H}. It is easy to see that CG(H) is a subgroup of G, and CG(G) = Z(G). Note that CG(H) ⊆ NG(H).

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