By Larry C. Grove

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Cs=l 4 The Sylow Theorems 21 Corollary. If a finite group G has a unique p-Sylow subgroup P for some prime p , then P 4G. 4 (The Third Sylow Theorem). If G is a finite group and p E if is a prime, then the number of distinct p-Sylow subgroups of G is congruent to 1 modulo p . Proof. Let P be p-Sylow and again let Y be the set of all G-conjugates of P . ,Yk. Note that { P ) is an orbit with just one element, say 9, = { P I . 2. 441 = l(mod p ) . 2. If P is a p-Sylow subgroup of G show that N,(N,(P)) NdP).

5. (1) Show that Z ( n u G d l= ) H,Z(G,). (2) Show that (G, x G2 x -.. x G,)’ = G; x G; x . - x GL. (3) Under what circumstances is G, x G, x ... x G, solvable? 7. , where Zi+,/Zi= Z(G/Zi) for all i. 12, that each Zi is normal in G , so the definition makes sense. If Z , = G for some n, then G is called a nilpotent group. Clearly every abelian group is nilpotent, since 2 , = Z ( G ) = G. The symmetric groups S, are not nilpotent if n 2 3 since Z ( S , ) = 1 and hence all zi = 1. 1. If G is a finite p-group, then G is nilpotent.

Note that if 0 is a k-cycle, then 101 = k. If a = a l o 2... a,, disjoint, with ai a k,-cycle, then 101 is the least common multiple of k,, k,, . 5). The inverse of a k-cycle is easily obtained by writing the entries in the cycle in reverse order. For example, if a = (12345),then a - l = (54321). A 2-cycle (ab) is called a transposition. , : : : :: ( 1 2 3 . . (13)(12). 1 every permutation can be written as a product of transpositions. We say that a E S,,is euen if it is possible to write a as a product 3 The Symmetric and Alternating Groups 17 of an even number of transpositions; otherwise we say that o is odd.