# Let $H$ be a subgroup of a group $G$ and suppose that $g_1,g_2 ∈ G$. Prove that the following conditions are equivalent:

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.

• Have you started with anything? What have you tried to get from (a) to (b)? – amsmath Mar 28 '19 at 1:01

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).
• @Claire you should make an equivalence between (e) and (a). Can you see that if you assume that $g_1^{-1}g_2 \in H$ then also $g_2^{-1}g_1 \in H$ as well? Many of the conditions above are symmetric, and will help you proving (d) implies (c) and (c) implies (b) for instance. – Mariah Mar 28 '19 at 2:06