15.2.2. Characterization for Subgroups

So, let us start with the definition of a subgroup. So, imagine you are given an abstract group $G$ with operation $\circ$, it may or may not be finite. And imagine I take a subset $H$ of the set $G$. Again it may or may not be finite. Of course, if $G$ is finite, any subset will be finite, but if $G$ is infinite then I may take a finite subset or infinite subset. Now, if the subset $H$ with the same operation $\circ$ satisfies the group axioms namely $\mathcal{A}_1, \mathcal{A}_2, \mathcal{A}_3, \mathcal{A}_4$ then I will call $H$ along with the operation $\circ$ to be a subgroup of the original group.

A subgroup is a

Think of a subgroup as a smaller team within a larger organization. If the whole organization represents the original group, the smaller team still needs to follow the same rules and structure (operation) as the organization.

An example of a subgroup will be the following. So, let group $G$ be the set of real numbers with the operation of integer addition then if I take the set of integers then that will be of course a subset of the real numbers. So, my real numbers was the bigger set and my integers is a subset of the set of real numbers and I take the same operation plus here. So, it is easy to see that the set of integers is indeed a group, it satisfies the group axioms with respect to integer addition and hence, I can say that this is a subgroup of the group of real numbers within integer addition.

In this example, real numbers and integers were considered. The integers are a subset of real numbers that also satisfy group properties under addition. Thus, the integers form a subgroup of real numbers, meaning all the integer operations will yield other integers.

Imagine a large library (real numbers) that contains books of different categories. If we consider only the books on mathematics (integers), it's like forming a smaller library that still operates under the rules of the larger library.

So now, an interesting question is, imagine you are given an abstract group and now, I give you a subset, how do I check whether it is a subgroup or not? There are 2 options, option one that you manually check whether all the group axioms are satisfied for the subset $H$ that you are given. But that is not what we will prefer because if my subset $H$ is very large then it might become very difficult to verify whether all the group axioms are satisfied or not.

To determine if a subset is a subgroup, checking all group conditions manually can be tedious especially for large subsets. Instead, we seek a more straightforward way to verify if it meets subgroup criteria.

It's like checking if a team of players adheres to game rules. Instead of observing every single play, you could just look for two essential rules — like having enough players and ensuring teamwork.

Instead, what we are looking now, here, is the following. We are looking for a characterization, some kind of condition which should be sufficient to check and declare whether the given subset $H$ satisfies the group axioms or not with respect to the operation. So here is a very interesting characterization for subgroups. So, you are given a subset $H$, of course, a subset $H$ has to be non-empty because if it is empty, it can never be a group because you need the identity element to be present at least in your group.

We can simplify the verification process for a subset $H$ by focusing on just two key criteria: non-emptiness and the closure property (the operation results in elements still within $H$), along with the existence of inverses. These criteria help simplify checking for subgroup status.

Think of a recipe: as long as you have the main ingredients (non-empty) and they can create delicious dishes (closure), it will lead to a successful meal (subgroup).

So, imagine you are given a non-empty subset, the characterization is the following. You just verify whether the two properties $\mathcal{C}$ and $\mathcal{I}$ are satisfied. And if they are satisfied then $H$ indeed constitutes a subgroup of the original group. Note that I am using the multiplicative notation here. What are these two conditions? The condition $\mathcal{C}$ demands that, the closure property should be satisfied.

The closure property means that for all elements $x, y$ in subset $H$, the operation $x \circ y$ must also produce an element in $H$. The inverse property states that for every element in $H$, there exists an inverse also in $H$. If both these hold, then the group axioms are satisfied.

Imagine a closed circle for a dance floor. Everyone can only dance (the operation) with others already on the floor, and every dancer has a partner (inverse) who can return them home.

So now, an interesting corollary here, is the following. The corollary says that, if your original group $G$ is finite then no need to check even for the second axiom; just check whether the first axiom is satisfied or not. Namely, just check whether the closure property is satisfied or not in your subset $H$. If the closure property is satisfied in the subset $H$ that automatically guarantees you that all the remaining group axioms are also satisfied in your subset.

For finite groups, you only need to verify that the closure property holds for a subset to conclude it is a subgroup. If $H$ is closed under the operation, it implies all other subgroup properties follow from it.

If you're playing board games (finite group) and the game's fundamental rule is just to not break the pieces (closure), you don't have to worry about every single rule as long as that one is followed.

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