This is Daily Work 15.

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Let \(H\) be a subgroup of \(G\) and suppose \(a,b\in G\) are elements of \(G\).

Which is not necessarily true?
Let \( H\) be a subgroup of \(G\), and suppose \(a\in G\) is an element.

Which of the following sets is not guaranteed to be a subgroup of \(G\)?
Let \(H_1,H_2\) be two distinct subgroups of a group \(G\), and let \(a \in G\) be an element.

Under what circumstances can we have \( aH_1 = H_2 \), that is, when can a left coset of one subgroup be equal to a different subgroup?

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