 Another important idea in abstract algebra is the notion of a subgroup, and this is pretty straightforward. So suppose I have some set G with operation star, and let this set be a group, and I'm going to take H to be a subset of G. If H with operation star is itself a group, then we say that H is a subgroup of G. And we also use this same notation. So one warning moving forward is that this notation we ordinarily would read as H is a subset of G. But now that we're talking about groups, we can now read this as H is a subgroup of G. And the difference is that sets don't have operations. Groups do have specific operations associated with them. So we could prove or disprove that H is a subgroup by showing that H is a group. Again, that's what the requirement says. If H with operation star is a group, then it's a subgroup. However, as mathematicians, we like to be as efficient as possible in our analysis. And so we know that since H is a subset of G and G is a group, then we inherit associativity. And the only things we have to verify are going to be closure. If I have two things in H, I want to make sure that A star B is also in H. I need to verify that the inverse, well we know that A inverse exists in G because G is a group. But I want to make sure that it exists in H because I want to say that H is going to be a group. So we need to verify that the inverse exists in H and we need to have the identity. Again, we know that the identity exists in our group G, so we need to make sure that our identity exists in our group H. Well, actually we can be a little bit more efficient. As long as we verify closure and inverse, we know that the inverse of anything is in H. And the product A star A inverse by closure has to be in H as well. And this is the identity. So once we've verified closure and inverse, we get identity for free. And so if we already verify these two properties, then H is going to be a subgroup. So for example, let's take the set of 2 by 2 matrices under matrix addition and let D be the set of 2 by 2 diagonal matrices. And let's prove or disprove D with the operation of addition is going to be a subgroup. So M with the operation of addition is a group, prove it. And so we just need to verify the closure and inverse properties for D. Well, let's take a note. If I have 2 by 2 matrices in D, then A plus B is going to be a diagonal matrix. So my sum is in D. I have closure. If A is in D, then minus A will be the additive inverse. And this is also going to be a diagonal matrix. So minus A is also going to be in D. So I have closure. I have the inverse. And that's all I need to verify that my subset is in fact a subgroup.