68 Chapter 1. Groups I Lastly, ϕ is a surjection: if y O(x), then y = gx for some g G, and so y = ϕ(gGx). In Example 1.102, D8 acting on the four corners of a square, we saw that |O(v0)| = 4, |Gv0 | = 2, and [G : Gv0 ] = 8/2 = 4. In Example 1.101, G = α Sn acting on X = {1, 2,...,n}, we saw that if α = β1 · · · βt is the complete factorization into disjoint cycles and occurs in the rj-cycle βj, then rj = |O( )|. Theorem 1.107 says that rj is a divisor of the order k of α (but Theorem 1.27 tells us more: k is the lcm of the lengths of the cycles occurring in the factorization). Corollary 1.108. If a finite group G acts on a set X, then the number of elements in any orbit is a divisor of |G|. Proof. This follows at once from Lagrange’s Theorem. Table 1 on page 10 displays the number of permutations in S4 of each cycle structure these numbers are 1, 6, 8, 6, 3. Note that each of these numbers is a divisor of |S4| = 24. Table 2 on page 10 shows that the corresponding numbers for S5 are 1, 10, 20, 30, 24, 20, and 15, and these are all divisors of |S5| = 120. We now recognize these subsets as being conjugacy classes, and the next corollary explains why these numbers divide the group order. Corollary 1.109. If x lies in a finite group G, then the number of conjugates of x is the index of its centralizer: |xG| = [G : CG(x)], and hence it is a divisor of |G|. Proof. As in Example 1.103, the orbit of x is its conjugacy class xG, and the stabilizer Gx is the centralizer CG(x). Proposition 1.110. If H is a subgroup of a finite group G, then the number of conjugates of H in G is [G : NG(H)]. Proof. As in Example 1.104, the orbit of H is the family of all its conjugates, and the stabilizer is its normalizer NG(H). There are some interesting applications of group actions to Combinatorics in the next section, but let us first apply group actions to Group Theory. When we began classifying groups of order 6, it would have been helpful to be able to assert that any such group has an element of order 3 (we were able to use an earlier exercise to assert the existence of an element of order 2). We now prove that if p is a prime divisor of |G|, where G is a finite group, then G contains an element of order p (Proposition 1.84 proved the special case of this when G is abelian). Theorem 1.111 (Cauchy). If G is a finite group whose order is divisible by a prime p, then G contains an element of order p.
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