I know of only one group, ##A_4## of order 12 which does not have a subgroup with order dividing the group size. In this case, a subgroup of size ##6##.(adsbygoogle = window.adsbygoogle || []).push({});

What property of a group causes this? Would I expect to find other examples only in non-abelian groups or are there abelian groups which do not have subgroup orders dividing the group size?

Surely there must be some reason for this anomaly right? Another words, if I looked at say 500 groups of any kind which do not contain subgroups dividing the group size, there would be no defining quality shared by all of these groups which is causing this property?

What other symmetric groups do not contain further subgroups dividing the subgroup size?

I suppose I can always number-crunch it to death and probably find a few . . . 500, not so sure. Ok, here's the task: Find all subgroups of ##S_n## which do not contain subgroups dividing the parent group order. Just how far could I numerically compute that at 2.2 GHz? 20? Any algebraic methods to find them?

Thanks,

Jack

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# Converse of Lagrange's Theorem for groups

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