Abstract
We make use of the primitive permutation action of degree 120 of the automorphism group of L 2 ( 16 ) , the special linear group of degree 2 over the field of order 16, to give an elementary definition of the automorphism group of the third Janko group J 3 and to prove its existence. This definition enables us to construct the 6156-point graph which is preserved by J 3 : 2 , to obtain the order of J 3 and to prove its simplicity. A presentation for J 3 : 2 follows from our definition, which also provides a concise notation for the elements of the group. We use this notation to give a representative of each of the conjugacy classes of J 3 : 2 .
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