Let $H$ be a subgroup of a group $G$ and suppose that $g_1,g_2 ∈ G$. Prove that the following conditions are equivalent:
(a) $g_1H = g_2H$,
(b) $Hg_1^{-1}=Hg_2^{-1}$,
(c) $g_1H \subset g_2H$,
(d) $g_2 \in g_1H$,
(e) $g_1^{-1}g_2 \in H$.
I'm beyond confused with this problem, all I know is that to prove the conditions are equivalent, I need to show that (a) implies (b), (b) implies (c), (c) implies (d), (d) implies (e), and (e) implies (a).
I have now gotten answers for (a) implying (e) and (e) implying (d). I'm overthinking all of this and am still confused about (d) implying (c) and (c) implying (b). When it comes to (b) implying (a), I thought I was getting somewhere but it doesn't seem to be working.
Best Answer
I'll get you started, maybe you're having trouble with the order of the clauses, but you should take the effort to do the rest yourself, as this is a basic question:
Assume $g_1H = g_2H$. Then $g_1^{-1}g_2H = g_1^{-1}g_1H = H$ so $g_1^{-1}g_2 \in H$. So (a) implies (e).
Assume (e). Then there is $h \in H$ such that $g_1^{-1}g_2 = h$. Thus $g_2 = g_1h\in g_1H$. So (e) implies (d).