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12.6 Further Topics on Simple Groups 387 12.6 Further Topics on Simple Groups

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12.6 Further Topics on Simple Groups 387 12.6 Further Topics on Simple Groups 12.6 Further Topics on Simple groups 387 12.6 Further Topics on Simple Groups This Web Section has three parts (a), (b) and (c). Part (a) gives a brief descriptions of the 56 (isomorphism classes of) simple groups of order less than 106, part (b) provides a second proof of the simplicity of the linear groups Ln(q), and part (c) discusses an ingenious method for constructing a version of the Steiner system S(5, 6, 12) from which several versions of S(4, 5, 11), the system for M11, can be computed. 12.6(a) Simple Groups of Order less than 106 The table below and the notes on the following five pages lists the basic facts concerning the non-Abelian simple groups of order less than 106. Further details are given in the Atlas (1985), note that some of the most interesting and important groups, for example the Mathieu group M24, have orders in excess of 108 and in many cases considerably more. Simple Order Prime Schur Outer Min Simple Order Prime Schur Outer Min group factor multi. auto. simple or group factor multi. auto. simple or count group group N-group count group group N-group ? A5 60 4 C2 C2 m-s L2(73) 194472 7 C2 C2 m-s ? 2 A6 360 6 C6 C2 N-g L2(79) 246480 8 C2 C2 N-g A7 2520 7 C6 C2 N-g L2(64) 262080 11 hei C6 N-g ? A8 20160 10 C2 C2 - L2(81) 265680 10 C2 C2 × C4 N-g A9 181440 12 C2 C2 - L2(83) 285852 6 C2 C2 m-s ? L2(4) 60 4 C2 C2 m-s L2(89) 352440 8 C2 C2 N-g ? L2(5) 60 4 C2 C2 m-s L2(97) 456288 9 C2 C2 m-s ? L2(7) 168 5 C2 C2 m-s L2(101) 515100 7 C2 C2 N-g ? 2 L2(9) 360 6 C6 C2 N-g L2(103) 546312 7 C2 C2 m-s L2(8) 504 6 C2 C3 m-s L2(107) 612468 7 C2 C2 m-s L2(11) 660 5 C2 C2 N-g L2(109) 647460 8 C2 C2 N-g L2(13) 1092 5 C2 C2 m-s L2(113) 721392 8 C2 C2 m-s L2(17) 2448 7 C2 C2 m-s L2(121) 885720 8 C2 C2 × C2 N-g L2(19) 3420 6 C2 C2 N-g L2(125) 976500 9 C2 C6 N-g ? L2(16) 4080 7 hei C4 N-g L3(2) 168 5 C2 C2 m-s L2(23) 6072 6 C2 C2 m-s L3(3) 5616 8 hei C2 m-s 2 2 L2(25) 7800 7 C2 C2 N-g L3(4) 20160 10 C3 ×C4 D6 - L2(27) 9828 7 C2 C6 m-s L3(5) 372000 10 hei C2 - ? L2(29) 12180 6 C2 C2 N-g L4(2) 20160 10 C2 C2 - L2(31) 14480 8 C2 C2 N-g U3(3) 6048 9 hei C2 N-g L2(37) 25308 6 C2 C2 m-s U3(4) 62400 10 hei C4 - L2(32) 32736 8 hei C5 m-s U3(5) 126000 10 C3 D3 - ? L2(41) 34440 7 C2 C2 N-g U4(2) 25920 11 C2 C2 - ? L2(43) 39732 6 C2 C2 m-s S4(3) 25920 11 C2 C2 - L2(47) 51888 7 C2 C2 m-s S4(4) 979200 13 hei C4 - 2 L2(49) 58800 9 C2 C2 N-g Sz(8) 29120 9 C2 × C2 C3 m-s L2(53) 74412 7 C2 C2 m-s M11 7920 8 hei hei N-g L2(59) 102660 6 C2 C2 N-g M12 95040 11 C2 C2 - L2(61) 113460 6 C2 C2 N-g M22 443520 12 C12 C2 - L2(67) 150348 6 C2 C2 m-s J1 175560 8 hei hei - L2(71) 178920 8 C2 C2 N-g J2 604800 13 C2 C2 - Table 1 – Non-Abelian Simple Groups with Order less than 106 388 Chapter 12 Simple Groups of Order less than 106 Notes. (a) The outer automorphism and Schur multiplier groups (Columns 4 and 5 above) are considered at the end of this subsection. Thompson (1968) defined an N-group, N-g, to be a group in which the normalisers of all of its non-neutral soluble subgroups are themselves soluble, and a group is called minimum simple, m-s, if all of its proper subgroups are soluble. As a devel- opment of his, and Feit’s, work on the odd order problem published in 1963, he was able to determine those simple groups that have the properties m-s and/or N-g. His Main Theorem is as follows: Each non-soluble N-group is isomorphic to a group G which satisfies: Inn H G Aut H where H is one of the following N-groups ≤ ≤ L (q) (q> 3), Sz(22n+1) (n 1), A , L (3), U (3) and M , 2 ≥ 7 3 3 11 see Sections 12.3 and 12.4, and (c) below (for the Suzuki groups Sz). As a corollary to this theorem, Thompson was also able to list the minimal simple groups: If p denotes an odd prime they are p p p L2(2 ), L2(3 ), L2(p), Sz(2 ), and L3(3), where in the third case p > 3 and p 3 or 7 (mod10). Column 6 of the table on the previous page lists this data≡ for the groups under discussion. As an exercise the reader should consider why the second list above is smaller than the first, see Problem 12.16W. Note that both A5 and A6 occur in the second list, see (b) below. (b) The stars (?) in Table 1 overleaf indicate that the isomorphism class of the group listed contains more than one group in the table; we have A L (4) L (5) 5 ' 2 ' 2 A L (9) 6 ' 2 A L (2) 8 ' 4 L (7) L (2) 2 ' 3 U (2) S (3), 4 ' 4 see Problems 6.16, 12.4, 12.6, 12.7 and 12.17, and the Atlas. The group A8 is not isomorphic to L3(4) even though these groups have the same order. For instance the group A8 contains elements of order 15 whilst L3(4) contains no element of order larger than 7, a number of other distinctions have been found; see Problem 12.13. Infinitely many pairs of non-isomorphic simple groups with the same order exist, but there are no triples. This is a consequence of the classification given by CFGS. (c) The Suzuki groups Sz(22n+1),n =1, 2,... were discovered by Muchio Suzuki in 1959. Later it was realised that they can also be treated as the (twisted) Chevalley groups 2B (22n+1). One definition is as a set of 4 4 2 × matrices over F22n+1 which ‘preserve’ a collection of vectors whose coordinates satisfy a special quadratic condition; we shall consider these groups in Web Section 14.3. They are the only non-Abelian simple groups whose orders are not divisible by 3, another result due to Thompson. 12.6 Further Topics on Simple groups 389 (d) Below we give brief descriptions of the groups in the table not so far mentioned in the text, or discussed in Web Chapter 14; they are S4(4),M22,J1 and J2. The unitary groups U3(4) to U4(2) have similar defi- nitions to that given for U3(3) on pages 265 to 267. The following data gives a partial list of their maximal subgroups, full details can be found in the Atlas 2 . U3(4) : A5 C5, S3 o C5 and F13,3; U3(5) : A7, C2S5 and M10 which × 4 is isomorphic to the group D defined on page 268; and U4(2) : A5 o C2 , S6 3 and S4 o C3 . One definition of the Mathieu group M12 was given on page 268 and its connection with the Steiner system S(5, 6, 12) will be considered on page 395. With M24 it is the only group that is 5-transitive if we exclude the symmetric and alternating cases. Several definitions are given in the Atlas including the following presentation M a,b,c a11 = b2 = c2 = (ab)3 = (ac)3 = (bc)10 = a2(bc)2a(bc)8 = e . 12 ' h | i The maximal subgroups are isomorphic copies of M11, an extension of T2 by 2 A6, C2S4 o C3 , C2 S5, an extension of an Extra Special group of order 32 by S , D o C2, and×A S . 3 6 4 4 × 3 S4(4) – The symplectic group of order 4 defined over F4 The def- inition of the group Sn(q) is similar to that for the unitary group Un(q). As before we work over the field Fq2 , the dimension n must now be even. The hermitian form fixed by matrices in Un(q) is replaced by an ‘alternating form’ of the type x y + x y + + x y x y x y x y 1 m+1 2 m+2 ··· m 2m − m+1 1 − m+2 2 −···− 2m m and matrices in S2m(q) where n = 2m fix a form of this type. If we have A B S (q) where A, B, C and D are m m, then C D ∈ 2m × A>C = C>A, B>D = D>B, A>D C>B = I , − 2m where denotes the transpose. All matrices have determinant 1, and S2m(q) is obtained> by factoring out the scalar matrices as in the linear and unitary cases, the centre has order (q 1, 2) in this case.
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