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I am studying group theory. I understand that normal subgroups are important, but I don't know why subgroups are also important that there are many theorems about them (like the Sylow theorems). They might be interesting groups themselves, but other than that, may I wonder what are some uses of them? If there are subgroups of a group $G$, do they tell us information about the group structure of $G$?

Thank you.

HallaSurvivor
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Coco
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  • So you are saying there is nothing of value in studying, e.g., the subgroup lattice of a (nonabelian) simple group? – user10354138 Sep 27 '23 at 06:02
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    @user10354138 Sorry I don't mean this, I know they must be important for a reason so I want to know why. – Coco Sep 27 '23 at 06:24
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    A very baby example: Suppose you want to know if there are any nontrivial homomorphism $G_1\to G_2$ (e.g., if you have a natural $G_2$-action but not $G_1$ on some $X$ when you want to construct one). If you don't know the subgroups of $G_2$ how could you tell? Normal subgroups only solve first half of the epi-monic factorization $G_1\twoheadrightarrow G\hookrightarrow G_2$, namely $G_1\twoheadrightarrow G_1/K=G$. The second half, $G\hookrightarrow G_2$ is of course whether this $G$ occurs as a subgroup of $G_2$. – user10354138 Sep 27 '23 at 06:40
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    In my opinion, this post is too broad and opinion based. – user264745 Sep 27 '23 at 11:29
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    The question is indeed very broad, and for integers one could formulate it like "I understand that integers are important. But why are divisors of integers important?" Again the answer would be very broad, ranging from prime number theory to sum of divisors and so on. – Dietrich Burde Sep 27 '23 at 13:07
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    @MarianoSuárez-Álvarez: It is not weird at all! It is a very natural question for a beginner. Also in many textbooks/courses the answer is not clearly explained, and left for the student to figure out on their own. – testaccount Sep 28 '23 at 03:17
  • Indeed definitely legitimate and well exposed. I +1'd yesterday. – citadel Sep 28 '23 at 05:05

2 Answers2

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Subgroups are extremely important to the study of a group -- I really can't overstate that point. I could give dozens of reasons, but here's a few that you might find interesting:

If $G$ is complicated we can simplify it in one of two ways

  • we can look at a quotient of $G$ (which adds extra relationships between the group elements, for instance we might force things to commute)
  • we can look at a subgroup of $G$ (which restricts attention to certain elements, and ignores others. For instance we might look at only those elements which preserve some property)

Both of these directions help us better understand the structure of $G$ (and more importantly, its actions).

For an example of understanding the actions, it turns out that if $H \leq G$ and $H$ acts on a vector space $V$ (this is a very common situation), then we can build a $G$ action (called the induced representation) out of it. Oftentimes the $H$ actions are easier to study, and these induced representations help us get a handle on $G$ actions.

For an example of understanding the structure, say that $G$ is finite and of order $pqr$ for three primes (for instance, we might have $|G| = 3 \cdot 5 \cdot 19$). Then $G$ is solvable (see here for a proof). This tells us that $G$ is "almost abelian" in a certain sense, and so gives us useful structural information about $G$. Importantly, the only proof I know of this fact crucially uses the theory of sylow subgroups of $G$.

As a bonus fact, in galois theory we relate the subfields of some field to the subgroups of its "galois group". This is extremely useful for field theory, because fields are fairly complicated objects, and are often infinite $(\mathbb{Q}, \mathbb{Q}(\sqrt{2}), \mathbb{R}, \mathbb{C}$, etc). Understanding field extensions and subfields is crucial for solving polynomial equations (and, more abstractly, much of modern number theory), and the only reason we're able to understand these field extensions is because of their connection to the subgroups of a (finite) group. So we've turned the study of an infinite object into the study of a finite one!

I hope this helps ^_^

HallaSurvivor
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As you acknowledged already the importance of normal subgroups, let's start from this characterization of them: for $G$ finite, they are the $H\le G$ such that $g^{-1}Hg= H$ for every $g\in G$, whence $(H\supseteq) \bigcap_{g\in G}g^{-1}Hg=H$. At the other end of the range, we have the "uttermost non normal subgroups", namely the $H\le G$ such that $\bigcap_{g\in G}g^{-1}Hg=\{1\}$. From this property follows that the $G$-action by left multiplication on the left quotient set $G/H$ is faithful, meaning that $G$ sharperly embeds into $S_{[G:H]}(<S_{|G|})$: this yields a piece of information on the structure of $G$ based on the existence of a (definitely non normal) subgroup $H$ of $G$.

citadel
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