Primitive permutation groups of simple diagonal type |
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Authors: | L G Kovács |
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Institution: | (1) Australian National University, G.P.O. Box 4, 2601 Canberra, A.C.T., Australia |
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Abstract: | LetG be a finite primitive group such that there is only one minimal normal subgroupM inG, thisM is nonabelian and nonsimple, and a maximal normal subgroup ofM is regular. Further, letH be a point stabilizer inG. ThenH∩M is a (nonabelian simple) common complement inM to all the maximal normal subgroups ofM, and there is a natural identification ofM with a direct powerT
m of a nonabelian simple groupT in whichH∩M becomes the “diagonal” subgroup ofT
m: this is the origin of the title. It is proved here that two abstractly isomorphic primitive groups of this type are permutationally
isomorphic if (and obviously only if) their point stabilizers are abstractly isomorphic.
GivenT
m, consider first the set of all permutational isomorphism classes of those primitive groups of this type whose minimal normal
subgroups are abstractly isomorphic toT
m. Secondly, form the direct productS
m×OutT of the symmetric group of degreem and the outer automorphism group ofT (so OutT=AutT/InnT), and consider the set of the conjugacy classes of those subgroups inS
m×OutT whose projections inS
m are primitive. The second result of the paper is that there is a bijection between these two sets.
The third issue discussed concerns the number of distinct permutational isomorphism classes of groups of this type, which
can fall into a single abstract isomorphism class. |
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Keywords: | |
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