Let and be subgroups of group . Which of the following is False?
Pick an option to check yourself, or open the answer below.
Show answer & explanation
Correct answer: A. H1.H2 = {h1*h2 : h1 ∈ H1 and h2 ∈ H2} is a subgroup.
Explanation
For two arbitrary subgroups H1 and H2 of a group G, the product set H1H2 is itself a subgroup only when H1H2 = H2H1, which need not hold in a non-abelian group, so statement 1 is False in general. If both H1 and H2 are normal subgroups, their product H1H2 is always a subgroup (indeed a normal subgroup), so statement 2 is True. The intersection of any two subgroups is always a subgroup of G, so the intersection statements are True. Hence the false statement is option 1.More Discrete Structures and Optimization questions
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