If H and K are subgroups of G, thena)HK is a subgroup but H∩K may not be a subgroup.b)H ∩ K is a subgroup but HK may not be a subgroup.c)Either (a) or (b).d)None of the above.Correct answer is option 'B'. Can you explain this answer?

Question

If H and K are subgroups of G, thena)HK is a subgroup but H&cap;K may not be a subgroup.b)H &cap; K is a subgroup but HK may not be a subgroup.c)Either (a) or (b).d)None of the above.Correct answer is option 'B'. Can you explain this answer?

Answer

In general, the product of two subgroups H and K of a group G, denoted by HK, is not necessarily a subgroup of G. In order for HK to be a subgroup, certain conditions need to be satisfied, such as H and K being normal subgroups or G being an abelian group.

However, we can say that the intersection of H and K, denoted by H ∩ K, is always a subgroup of G. This is because the intersection of any collection of subgroups is itself a subgroup.

To clarify the statement you provided:
a) If H and K are subgroups of G, then HK may or may not be a subgroup of G.
b) If H and K are subgroups of G, then the intersection H ∩ K is always a subgroup of G.

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