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If the order of a group is pqr with p q r primes, then there exist three elements A B C with order p q r. Is it possible that the order of the subgroup generated by A and B has the order of pqr ?

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up vote 5 down vote accepted

Yes. Consider the group of 2x2 matrices whose entries come from the field of 7 elements. Take A = [ -1, 0 ; 0, 1 ], B = [ 2, 1 ; 0, 1 ], C = [ 1, 1 ; 0, 1 ]. Then A,B,C generate a group of order 42=2*3*7 called Hol(7) = AGL(1,7). A has order 2, B has order 3, C has order 7, but the subgroup generated by A and B also has order 42.

This is basically the same trick used to generate the the non-abelian group of order 6 using two elements of order 2. You can't take them from the same Sylow 2-subgroup, but if you choose from two Sylow 2-subgroups things work. For p=2, q=3, r=7, we cannot choose two from the same Hall {p,q}-subgroup, lest we get a subgroup of order pq, so we choose from two different Hall 6-subgroups, and generate the whole group.

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