 Welcome back to our lecture series, Math 4230, abstract algebra two for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misalign. In lecture one of this series, we introduced the idea of a group action, and we provided some very important properties and examples of group actions. In this very short first video for lecture two, I like to give us a couple more examples of group actions, just to get an add to our list here. Let's suppose that we have a G set X, that is, there's already an action of the group G on the set X. If X is a G set, then the power set of X is also a G set in a very, very natural way. So that is to say, we have some well-defined action, G acts on little sets X, this is equal to some element Y right here. If we take any subset of X, because these are the typical elements of the power set, then we can define the action of G on the set. And we're just gonna do this element-wise. So G dot S is then gonna be the collection of all of the elements of the form G dot little S, where little S is allowed to vary over all of the elements of the set there. And so to prove that this is a group action, we'd have to check that the identity axiom holds. That is to say, if the identity element of the group acts on a set, you get back to the original set, we also have to show compatibility, that if we have two group elements, G and H, if you act on the set first by H, then by G, that's the same thing as acting by the element G times H. And these two axioms are inherited from the axioms that X is a G set. So I'm actually gonna leave it as an exercise to the viewer to prove that the power set of a G set is itself a G set. So we can make new G sets from other ones, right? Another important example is if we have a group G, and we take the set of all subgroups of G, okay? This is very similar to the last example, in which case we aren't taking the power set, but we're taking a very important subset there. Because as we discovered previously in a previous video, G acts on itself, it acts on G via conjugation, right? What is conjugation again? So we have G dot X, this is equal to GXG inverse, like so. So a group acts on itself. So taking the previous example, since G acts upon G, we can then extend G to act upon the power set of G. That is, if we take sets, subsets of the group, we can conjugate those subsets, and that would be a group action. But we don't want the whole power set, we actually wanna restrict it to a very important subset of the power set, which is the collection of subgroups. So X is the set of all subgroups of G. Then this power set action does restrict to subgroups if we act by conjugation. That is to say G dot H, where G is an arbitrary element of big G, and H is just an arbitrary subgroup. G acts upon H by conjugation. We can take the conjugate subgroup, GHG inverse. This would be the collection of all elements of the form GHG inverse. It's important to point out that the conjugate of a subgroup is in fact a subgroup itself. It contains the identity because GEG inverse is just the identity, right? It's closed under multiplication because the product of two conjugates. So you have X and Y there. This simplifies to be GXYG inverse. This belongs there. You have inversion as well because put that back on there. If you take the inverse of this element, this becomes GX inverse, G inverse. So it's important to know that the conjugate of a subgroup is in fact a subgroup. So this action is well-defined. That is it produces a subgroup. The compatibility and the identity axioms, I'll leave it to the viewer also to provide those details as well. And so one other comment I wanna say before we close this video here is that the set of subgroups, the set of subgroups of a group G is clearly a subset of the power set of G. And because G acts upon itself by conjugation, that conjugation axiom can be extended to the power set and then we restricted it to the set of subgroups right here. So this right here, so this power set of G is in fact a G set. It means that G acts upon it. This is a G set inside of a G set. So this is what we call a G subset. So it's kind of like a subgroup. What's a subgroup? A subgroup is a group inside of another group for which the operation is just restriction from the overgroup. A G subset is the same basic idea that if you have a G set, if you restrict to a subset which is also closed under the group action, you get a G subset. And forming G subsets just comes down to taking unions of orbits. If you take all of the orbits in any set, if you just take some subset of orbits that gives you a G subset. And so the collection of subgroups is a conjugation G subset of the power set.