I'm new to group theory, and trying to understand some basic concepts about cosets.
I found the following 5 lemmas and their proof was left "as an exercise for the reader" but I can't figure them out.
Let $H$ be a subgroup of $G$ and $gH$ be a left coset with $g$ as its representative. I want to show the following 5 lemmas are equivalent:
- $g_1H = g_2H$
- $Hg_1^{-1} = Hg_2^{-1}$
- $g_1H \subseteq g_2H$
- $g_2 \in g_1H$
- $g_1^{-1}g_2 \in H$
In particular, I want to show 1) implies 2) implies 3) implies 4) implies 5) implies 1) again.
I don't know how to get from 1) to 2). But it's obvious that 1) implies 3) but the other way around is nonobvious.
1) implies 4) since the identity is in $H$, so $g_2 \in g_2H \in g_1H$
4) implies 5) by left multiplication with $g_1$. Again, 5) to 1) baffles me since I can't get from the $\in$ to an $=$.
Can someone suggest approaches for showing these equivalences?