Order of product of subgroups

Algebra Level 3

Let \(G\) be a group and let \(H\) and \(K\) be two subgroups of \(G\).

If both \(H\) and \(K\) have \(12\) elements, which of the following numbers cannot be the the number of elements in the set \(HK\) ?


Notation:

  • \(HK = \{hk\mid h\in H~\textrm{and}~k\in K\}\)
Problem Source: TIFR Entrance Exam
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