Order of product of subgroups

Algebra Level 3

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

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

Notation:

  • H K = { h k ∣ h ∈ H and k ∈ K } HK = \{hk\mid h\in H~\textrm{and}~k\in K\}
Problem Source: TIFR Entrance Exam
48 60 36 72

Prerona Chatterjee by Prerona Chatterjee , 28, India

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1 solution

A group of order twelve has 4 possible subgroups. 6, 4, 3, 2. If H H and K K were to share the same subgroup then it would over lap. Therefore H K HK can only have 6 × 12 , 4 × 12 , 3 × 12 , 2 × 12 6\times12,\;4\times12,\;3\times12,2\times12 elements. Thus 60 \boxed{60} is the odd one out.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

kernel and factor groups, i suck at this

Kaye Pajaron - 6 years, 7 months ago

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