Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 ?

share|improve this question

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.