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# JHolder - 16 Notes on Jordan-Hlder o Denition 16.1 A normal...

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16 Notes on Jordan-H¨older Definition 16.1. A normal series of a group G is a sequence of subgroups: G = H 0 H 1 H 2 ≥ · · · ≥ H m = 1 so that each subgroup is normal in the previous one ( H i E H i - 1 ). The quotient groups H i - 1 /H i is called a subquotients . A refinement of a normal series is another normal series obtained by adding extra terms. Example 16.2. Here are some examples of normal series. 1. S n > A n > 1 is a normal series for G = S n for any n 1. The associated subquotients are S n /A n = Z / 2 and A n / 1 = A n . 2. For n = 4 this normal series has a refinement: S 4 > A 4 > K > 1 The subquotients are (a) S 4 /A 4 = Z / 2 (b) A 4 /K = Z 3 (Any group of order 3 is cyclic.) (c) K = Z / 2 × Z / 2 3. A normal series for the dihedral group D 8 = h a, b | a 2 , b 4 , abab i is given by D 8 > h b i > b 2 fi > 1 . All three subquotients are cyclic of order 2. 4. G = GL n ( R ) has a normal series GL n ( R ) SL n ( R ) Z ( GL n ( R )) 1 The subquotients are: (a) GL n ( R ) /SL n ( R ) = R × (b) SL n ( R ) /Z ( GL n ( R )) = PSL n ( R ) by definition of the projective unimodular group PSL n ( R ). (c) Z ( GL n ( R )) = R × since only scalar multiples of the identity are central. 5. (A stupid example) Any group G has this normal series: G B 1 . If G is a simple group, there is the only normal series for G . 1

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Definition 16.3. A composition series is a normal series G = G 0 > G 1 > · · · > G n = 1 so that G i is a maximal proper normal subgroup of G i - 1 for 1 i n . The subquotients G i - 1 /G i are simple groups called the composition factors of G . In Example 16.2, (1), for n 5 and (3) are composition series but (2) is not because K is not simple. A composition series for S 4 is given by refining (3) by adding one more term: H = the cyclic group generated by (12)(34): S 4 > A 4 > K > H > 1 The composition factors are Z / 2 , Z / 3 , Z / 2 , Z / 2. Example (4) GL n ( R ) does
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JHolder - 16 Notes on Jordan-Hlder o Denition 16.1 A normal...

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