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PROPERTIES OF GROUPS HAVING A SMALL
PROPERTIES OF A MINIMAL PAIR
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2-power 2aisi 3-element assume Burnside's theorem characteristic subgroup composition factors conclude that G Conjecture holds Conjecture is true contain H contains elements cycle structure denote distinct divides G divisor double coset element of H elements of order elements x e finite group follows that G Frobenius Furthermore G is simple group G Hence index at least involutions kisidi klsldl Kpa(G l(mod Let G Let H minimal pair Mn(G modulo Mp(G Mpa(G multiple n a positive integer N(HX normal subgroup NQ(H number of elements number of subgroups Op(G p a prime p-group p-subgroups pa(kp pa(mp pairwise intersection particular permutation representation positive integer dividing proper subgroup Proposition 2.1 simple group simplicity of G solvable groups STEP subgroup of G subgroup of index subgroups of order Syl G Sylo(G Sylow 3-subgroup Sylow's theorem whence G x e G