My apologies, this must seem like a trivial question, but this image below appeared on Socratica on youtube and I wanted to clarify my thoughts on cosets and subgroups. From the image below, I can see that the left coset $g_1$h is the composition of all elements in the subgroup h in H with an element $g_1$ from outside the set.
I think of this as the elements in H are being "translated" out of the subgroup H and into a different object (a coset altogether).
My question is since the left coset does not contain the identity element, it is therefore not a subgroup? Am I right in this reasoning?
Here is the video – the system doesn't appear to let me load images with my reputation.
See 4 min 55 sec
Best Answer
You are right. A coset is not necessarily a subgroup and one of the features is what you said that it may not contain the identity. In general you can easily see in any examples that it may not be closed under the group operation.