Trouble understanding the concept of a subgroup generated by a subset $S$, $\langle S \rangle$

Currently self-studying Abstract Algebra through Goodmans "Algebra: Abstract and Concrete" and having a bit of a hangup with the concept of a subgroup generated by a subset $S$.

The books definitions of $\langle S \rangle$ and $\langle a \rangle$ are as follows:

For a subset $S \subseteq G$ of a group $G$ the smallest subgroup of $G$ that contains the set $S$ is $\langle S \rangle$. A constructive view of $\langle S \rangle$ is that it consists of all possible products $g_{1}g_{2} \dots g_{n}$, where $g_{i} \in S$ or $g^{-1}_{i} \in S$. When $S$ = $\langle a \rangle$, a singleton, the subgroup generated by $S$ is denoted by $\langle a \rangle$

An alternative definition I have come across checking this website comes from the question here

For any element a of a group $G$ it is useful to think of $\langle a\rangle$ as the smallest subgroup of $G$ containing $a$. This notion can be extended to any collection $S$ of elements from a group $G$ by defining $\langle S\rangle$ as the subgroup of $G$ with the property that $\langle S\rangle$ contains $S$ and if $H$ is any subgroup of $G$ containing $S$, then $H$ also contains $\langle S\rangle$. Thus, $\langle S\rangle$ is the smallest subgroup of $G$ that contains $S$. The set $\langle S\rangle$ is called the subgroup generated by $S.$"

I think I understand the very basics of the definitions. That, if we take a subset $S$ of a group $G$, say the group of symmetries of the square $G = \{ e,r,r^{2},r^{3},a,b,c,d\}$ and the subset $S= \{ e,a,b\}$ then the smallest subgroup which contains this set would be $\{ e, a, r^{2}, b\}$. This is so, because this is the smallest of the subgroups that contains all elements of $S$ - The main group $G$ contains all elements but is larger and the subgroups $\{ e, a\}, \{ e, r^{2}\}$ and $\{ e, b\}$ are smaller but do not contain all the elements in the set. As well, looking at this from the constructive set point of view if we were to multiply the elements of the set we would get as products $e, a,b$ and $r^{2}$ since the product $ab = r^{2}$. Is this correct?

For a singleton, say $S = \{ c\}$ then $\langle c \rangle$ would be $\{ e,c\}$. Correct?

As well, the notion of the unique smallest and largest subgroup in a lattice is giving me some trouble. The definition the book gives for each is:

Given $2$ Subgroups $A$ and $B$ of $G$

1. There is a unique smallest subgroup $C$ such that: $ C \supseteq A$ and $C \supseteq B$. In fact $C = \langle A \cup B \rangle$
2. There is a unique largest subgroup $D$ such that: $ D \subseteq A$ and $D \subseteq B$. In fact $D = A \cap B$

So given the example group above of the symmetries of the square here if we take. $A= \{ e,a,r^{2},b\}$ and $B = \{ e,d,r^{2},c\}$ then we would say that $C = \langle A \cup B \rangle$ = $\langle\{ e,a,b,r^{2},c,d\}\rangle= G$ since $G$ is the smallest subgroup that contains all of these elements. Correct?

$D= A \cap B= \{e,r^{2}\}.$ This makes sense, I believe, since it is the largest subgroup that contains the elements that are in both $A$ and $B$. Correct?

I guess I'm having a bit of trouble gaining some intuition on this concept. It feels a bit clearer to me now that I've sit down, collected all the information and wrote it out. As well, not having anyone to check my understanding can lead to reinforcing wrong notions and a lack of clarity. I believe I was also getting confused by the same symbolism being used for the Cyclic Subgroup of $G$ generated by $a$, $\langle a \rangle = \{a^{k} : k \in \mathbb{Z}\}$.

I'm just looking for clarity on whether or not my understanding is correct and if anyone has any examples they would like to provide or have found particularly illuminating I would appreciate that. As my book doesn't provide any examples.

I apologize for the length of this post. I didn't see any other posts about this concept that were as basic on this site though so I'm hoping it will serve a purpose, for posterity, and help someone else who is/was as confused as I was.

Solution 1:

It helps to think of $\langle S\rangle$ as the subgroup consisting of all (finite) products of all elements of $S$ and their inverses, so that if $S=\{s_i\mid i\in I\}$ for some index set $I$, then

$$\langle S\rangle =\left\{ s_{i_1}^{\varepsilon_{i_1}}\dots s_{i_k}^{\varepsilon_{i_k}}\,\middle|\, i_j\in I, k\in\Bbb N, \varepsilon_{i_j}\in\{1,-1\}\right\}\cup\{ e\}.$$

In this light, then, the subgroup of the symmetries of a square, generated by $\{e,a,b\}$, is indeed $\{e,a,r^2,b\}$ since, as you identified, $ab=r^2$.

Since $c^2=e$, we have $\langle c\rangle=\{ e, c\}$.

Yes, $C$ is as you described. Moreover, it is indeed the whole group $G$.

Yes, that is the intersection.