By Chen G.

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4); similarly He,2 < 1(e,4. The other possibilities for CD are treated similarly. A. 5. A, one may substitute keroe'r) for kerc(7r). 5 that C lie:, in the kernel of l' if and only if G preserves the two families of totally singubr ¥--spaces in the O~( q )-geomctry. Also observe that in all but the first two rows of the table, will arise. This concludes Example 6. H with the observation that the G-novelties which occur are always maximal in G, except in the follO"lving cases: (a) (b) n;::::; 5L,,(2), Ho of type GL I (2) I 5 n with 11.

V\le return to our previous example with n ;: : : 5L I4 (729) and Hn E Cs(D) of type GL::1(9) ... Fi~st of all, we saw in Example 2 that ker(7r) = (57, ¢3'i), and hence kCl'6(7r) = G n (07,¢3i) So.. 1), it remains to find 7r(Gi). Now if 7r(G) 9i Z7:ZI1l with 111 1 6, then 7r(G) is transitive and 7r(G i ) ~ Zm for each i. ntransitive. Write 17r(G)1 = 1n, with 111 1 6. 5) G has 1 + orbits on [Hn] A , with one orbit of size 1 and the others of size Tn. Thus if [H j ] is in the orbit of size I, we have if i = j if i f: j.

RI... ,I't" VJ "'II".... E. In the upper row corresponding to Cj = UCi(X), where X ranges between D and A, and between IT and if, as e 71 the 11111n1)('1' of types depends on the parity of q, as follows. When q is even, the number of types is just the number of ways of writing n in the form n = mt with t 2': 2 and (711, q) -:f. (2,2). before. Warning. Our definition of the collections Ci differs slightly from Aschbacher's original definition in [As 1 ). One difference is that we enlarge C1 so that it includes Aschbacher's in case L.

### A Characterization of Alternating Groups by the Set of Orders of Maximal Abelian Subgroups by Chen G.

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