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Theorem 92 Let G be a group of nite order m If g G then g m = e The examples we have presented raise a natural question If G is a group of order m and if k evenly divides m then does it follow that G has a subgroup of order k In general the answer is no You are requested in the exercises to provide a counterexample However, the following theorem of Sylow provides a positive answer in a large number of important instances Theorem 93 (Sylow) Let G be a group of nite order and let p be a prime Suppose that j is a positive integer and that p j evenly divides the order of G Then G has a subgroup of order p j Indeed, it has subgroups of all orders p , 0 j Proof of a Special Case of the Sylow Theorems: Let G be a nite group and suppose that the prime integer p divides the order n of G We shall show that G has a subgroup of order p First suppose that the integer m has the property that g m = e for every g G (we call m an exponent for G) Let b G, b = e, and let H be the cyclic group generated by b (that is, the collection of all powers of b) It can be checked that G/H forms a group Then of course bm = 1 hence m is an exponent for G/H Thus the order of G/H divides a power of m Because [G : e] = [G : H ] [H : e] we may conclude therefore that the order of G divides a power of m Since p divides n, there an element x G such that the period of x (that is, the minimal power of x that equals e) is divisible by p Let the period be ps for some integer s Then x s = e and x s has period p Thus x s generates a subgroup of order p That is what was to be shown We refer the interested reader to [LAN] or [HER] for a complete consideration of the Sylow theorems and their proofs We close with a few remarks about the concept of isomorphism of groups Consider rst an example You are familiar with the group Z2 It is a group of order 2 Now suppose we consider a group G with two elements: G = {e, m} The element e will be the group identity and the rules of multiplication are e m =m m e =m e e =e m m =e.

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You can check for yourself that, with this binary operation, G is indeed a group that is, it satis es the axioms for a group On a formal level, the group Z2 and the group G are different One has elements 0 mod 2 , 1 mod 2 and the other has elements e, m But in fact it turns out that they

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are the same group; they differ only in the sense that different names have been given to the elements To see this, write out the group law for Z2 : 0+1=1 1+0=1 0+0=0 1+1=0

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