PERSPECTIVITIESOne of the most fundamental results in the theory of projective planes is due to R. Baer: Let ~ be a projective plane and i a collineation of ~ of order 2, then either i is a perspectivity or planar.There are almost no general results dealing with collineation groups all of whose involutions are planar. Conversely, if a collineation group contains perspectivities of order 2, then a bunch of general theorems are available. Even if the restriction on the order of a perspectivity is dropped, still better results are known. A milestone among these is the following fundamental theorem due to Chr. Hering [25] : Let G be a finite collineation group acting strongly irreducibly on a projective plane ~. If G contains perspectivities, then with one exception G is an automorphism group of a non-abelian simple group.Geometriae Dedicata 13 (1982) 07-46. 0046-5755/82/0131-007506.00. Proof. The list of maximal subgroups can be found in [44] ; for the remaining statements see [31]. LEMMA 1.3. Let G be isomorphic with Mathieu's group M22. Then the following holds."(a) [GI = 27"32"5"7"11. (b) If M is a maximal subgroup of G, then M ~ La(4), E16 '2~5' E16"A6, AT, E 8 • L3(2), L2(11 ), or A6"2. (c) All involutions in G are conjugate and if t is an involution in G, then C~(t) = E16"24, a subgroup of the maximal subgroup E16'A6.Proof. The list of maximal subgroups can be found in [6], whereas for the remaining statements see [35].

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