The JordanâHÃ¶lder Theorem

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Hence, once we have determined one composition series for a (say, finite) group, then we have uniquely determined composition factors whichcanbe thought of as the ways of breaking our original group down into simple groups. This is analogous to a ‘prime factorisation’ for groups. To recognise when we have a composition series, we need to be able to recognise simple groups. We have already observed (Example 3.6) that cyclic groups of prime order are simple. It is not hard to show that these are all the abelian simple groups. The following was proved in MT4003. Theorem 4.7 Let n 5. ThenthealternatinggroupA n is simple. It is worth pointing out that much, much more is known. A mammoth effort by hundreds of mathematicians from the 1950s to the 1980s succeeded in classifying the finite simple groups. The complete proof runs to tens of thousands of pages of extremely complicated mathematics. More work is still being done to check, clarify and simplify the proof. Nevertheless, it is generally accepted that this Classification is correct, though when relying upon it a mathematician would normally state that he or she is doing so. Theorem 4.8 (Classification of Finite Simple Groups) Let G be a finite simple group. Then G is one of the following: (i) acyclicgroupofprimeorder; (ii) an alternating group A n where n 5; (iii) one of sixteen infinite families of groups of Lie type; (iv) one of twenty-six sporadic simple groups. The groups of Lie type are essentially ‘matrix-like’ groups which preserve geometric structures on vector spaces over finite fields. For example, the first (and most easily described) family is PSL n (q) = SL n(q) Z(SL n (q)) . To make it, first construct the group GL n (q) ofinvertiblen × n matrices with entries from the finite field F of order q. Next, take the subgroup of matrices of determinant 1: SL n (q) ={ A ∈ GL n (q) | det A =1}, the special linear group. Now,factorbythecentreofSL n (q), which consists of all scalar matrices of determinant 1: Z(SL n (q)) = { λI | λ ∈ F , λ n =1}. 47
The result is PSL n (q). It can be shown that PSL n (q) issimpleifandonlyif either n =2andq 4, or n 3. The construction of the other 15 families of groups of Lie type is similar but harder. The twenty-six sporadic groups are: Name Symbolic name Order Mathieu M 11 7920 Mathieu M 12 95 040 Janko J 1 175 560 Mathieu M 22 443 520 Janko J 2 604 800 Mathieu M 23 10 200 960 Higman–Sims HS 44 352 000 Janko J 3 50 232 960 Mathieu M 24 244 823 040 McLaughlin McL 898 128 000 Held He 4030387200 Rudvalis Ru 145 926 144 000 Suzuki Suz 448 345 497 600 O’Nan O’N 460 815 505 920 Conway Co 3 495 766 656 000 Conway Co 2 42 305 421 312 000 Fischer Fi 22 64 561 751 654 400 Harada–Norton HN 273 030 912 000 000 Lyons Ly 51 765 179 004 000 000 Thompson Th 90 745 943 887 872 000 Fischer Fi 23 4089470473293004800 Conway Co 1 4157776806543360000 Janko J 4 86 775 571 046 077 562 880 Fischer Fi ′ 24 1255205709190661721292800 Baby Monster B 4 154 781 481 226 426 191 177 580 544 000 000 Monster M see below |M| =808017424794512875886459904961710757005754368000000000 Unsurprisingly, we omit the proof of Theorem 4.8. We finish this chapter with a few examples. Example 4.9 Let G be a finite abelian group of order n. Write n = p r 1 1 pr 2 2 ...prs s where p 1 , p 2 ,...,p s are the distinct prime factors of n. If G = G 0 >G 1 >G 2 > ··· >G m = 1 48
Page 1 and 2: Chapter 4 The Jordan-Hölder Theore
Page 3: Proof: Let G be a finite group. We

The JordanâHÃ¶lder Theorem

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The JordanâHÃ¶lder Theorem