Subgroups Part 1

A subgroup is a subset of a group that with the restricted composition table inherited from the parent group satisfies the axioms of group theory as follows:
The subgroup’s restricted composition table must be closed and only contain elements of the subgroup.
The identity must be within the subgroup.
Associativity will be effortlessly inherited from the parent group.
For all elements within the subgroup, their inverse must also be in the subgroup.
In this video we explain this definition in detail and then go through some examples.
One of the examples we look at is the subgroups of the group of integers under addition. We find all the possible subgroups of this group and show how all but the trivial subgroup are isomorphic to the original group.

[@tejasplants]

Sir how do I know more about you.
I m so much suprised that you do maths so good and medicine subject too.

[@shadaanhassan5936]

Legendary❤

[@waynemv]

I can easily see why a group can only have one identity element. If i and j are both identity elements of the same group, we'd have both ij=i and ij=j, which is a contradiction.
If H is a subgroup of G, why does the identity element of G necessarily have to be the identity element of H? Might not possibly some other element, instead of e, play the same role of the identity element within H? Maybe elements of G are {e, a, b, c} and thus, ee=e, ae=ea=a, be=eb=b, and ce=ec=c; but elements of subgroup H might be {a, b} with aa=a and ab=ba=b? I don't yet see how this necessarily leads to any contradiction.

[@gianniskyriakou6577]

It becomes evident when you stop thinking of H as some 'distinct' group. It is simply just a subset of the group G. Then use the same logic for why you can't have a second identity element and it becomes clear.

[@MuffinsAPlenty]

@@gianniskyriakou6577 To further Giannis Kyriakou's point - it's not just that H is a subset of G. When we talk about subgroups, we require that H has the _same operation_ as G. In other words, the multiplication map μ : HxH → H is a _restriction_ of the multiplication map μ : GxG → G to the subset HxH of GxG. So it's _literally_ the same operation. Combined with the fact that inverses are _necessary_ by the axioms of a group, the identity has to be the same in H as it is in G.

[@qxtr5853]

I really like these videos, but it's a bit strange that often half of them is spent almost word for word restating something you've already covered. Thank you for your efforts anyway 👍

[@elliotnicholson5117]

Thank you for the feedback. In more recent videos I try to be more to the point.

[@bonbonpony]

Instead of repeating the same stuff over and over in every video, a better idea is just refer to that original video in the video that needs the information from it. Everyone can then just "follow the link" and see that other video if one doesn't have that knowledge yet.

[@ILoveMaths07]

I actually prefer the repetitions.

[@liamr7293]

ahaaaaaaaaaaa