“How often misused words generate misleading thoughts.”
―Herbert Spencer
SubsectionGenerational Wealth
Definition2.1.Generated Subgroup.
Given a group \(G\) and a subset \(S\) of \(G\text{,}\) the subgroup of \(G\) generated by \(S\), written \(\langle S \rangle\text{,}\) is the smallest subgroup of \(G\) containing \(S\text{.}\) In symbols,
\begin{equation*}
\langle S \rangle := \bigcap_{H \leq G, S \subseteq H} H.
\end{equation*}
The elements of \(S\) are known as generators, and \(S\) is called a generating set.
Definition2.2.Finitely Generated.
A group \(G\) is finitely generated provided that \(G=\langle S \rangle\text{,}\) where \(S\) is a finite set.
We can describe the elements of \(\langle S \rangle\) explicitly.
Lemma2.3.Elements of \(\igen S\).
For a subset \(S\) of \(G\text{,}\) the elements of \(\langle S \rangle\) can be described as:
\begin{equation*}
\langle S \rangle = \left\{x_1^{j_1} \cdots x_m^{j_m} \mid m \geq 0, j_1, \dots, j_m \in \mathbb{Z}\text{ and }x_1, \dots, x_m \in S \right\}.
\end{equation*}
Proof.
For notational simplicity, we let \(X= \left\{x_1^{j_1} \cdots x_m^{j_m} \mid m \geq 0, j_1, \dots, j_m \in \mathbb{Z}\text{ and }x_1, \dots, x_m \in S \right\}\text{.}\) I claim \(X = \igen S\text{.}\)
\((X\sse \igen S)\).
Let \(z\in X\text{.}\) Thus \(z = x_1^{j_1} \cdots x_m^{j_m}\text{,}\) where \(m \geq 0, j_1, \dots, j_m \in \mathbb{Z}\) and \(x_1, \dots, x_m \in S\text{.}\)
Since \(\langle S \rangle\) is a subgroup of \(G\) containing \(S\text{,}\) we see \(x_1, \dots, x_m\in \igen S\text{.}\) Additionally, given the group group structure of \(\langle S \rangle\) it is closed under products and inverses, and thus contains \(x_1^{j_1} \cdots x_m^{j_m}\text{.}\) Hence \(z\in \igen S\) and \(X\sse\igen S\)
\((\igen S\sse X)\).
I claim \(X\) is a subgroup of \(G\text{.}\)
First, notice \(S \ne \emptyset\text{,}\) since we allow \(m = 0\) and declare the empty product to be \(e_G\text{.}\)
Let \(x = x_1^{j_1} \cdots x_m^{j_m}\) and \(y = y_1^{i_1} \cdots y_n^{i_m}\) be arbitrary in \(X\text{.}\) Thus \(x_j, y_i\in S\) for all \(i,j\text{.}\) Observe:
As \(x_1^{j_1} \cdots x_m^{j_m} y_n^{-i_m} \cdots y_1^{-i_1}\in X\) as well, we see \(X\leq G\) by the one-step subgroup test.
Since \(\igen S\) is defined to be the intersection of all subgroups of \(G\) containing \(S\text{,}\) we have \(\igen S\sse X\text{,}\) completing the proof.
Lets look at some examples for context.
Example2.4.Generated Groups.
If \(G\) is a group, then \(S=G\) is a generating set of \(G\text{.}\)
The trivial group is generated by the empty set.
If \(S=\{x\}\) is a set with one element then we write \(\langle S \rangle=\) and we refer to this as the cyclic subgroup generated by\(x\text{.}\)
\(n\Z=\igen n\) for every \(n\in\Z\text{.}\)
\(\displaystyle \igen{r,s}=D_{2n}\)
\(A_n\) is the subgroup of \(S_n\) generated by all produts of \(2\)-cyclies (disjoint or not). In symbols,
Let \(p\ge 3\) be prime and \(r,r'\) be any two distinct reflections in \(D_{2p}\text{.}\) Then \(\igen{r,r'}=D_{2p}\text{.}\)
Exercise2.6.Generating \(\GL_n(\R)\).
For any integer \(n \geq 1\text{,}\) the set of column vectors \(e_i\) consisting of \(1\) in the \(i\th\) row and \(0\) elsewhere generate \(\GL_n(\R)\text{.}\) As you may recall from a linear algebra course, these are called the elementary matrices. (Bonus: what subgroup do the “type I” elementary matrices generate?)
We look now at generating \(S_n\text{;}\) the following lemma will be helpful.
Lemma2.7.
For \(\sigma\in S_n\) and distinct intgers \(i_1, \dots, i_p\) we have
(Note that the right-hand cycle is a cycle since \(\sigma\) is one-to-one.)
Proof.
To prove this, evaluate both sides at \(\sigma(i_t)\) for any \(t\) and observe that one gets \(\sigma(i_{t+1})\) (with the supscript taken modulo \(p\)) both times. This proves they agree on the set \(\sigma(\{i_1, \dots, i_p\})\text{.}\) If \(j\) is not in this set, then \((i_1 \, i_2 \, \cdots i_p)\) fixes \(\sigma^{-1}(j)\) so the left-hand side fixes \(j\text{.}\) So does the right, since \(\sigma^{-1}(j) \notin \{\sigma(i_1), \dots, \sigma(i_p)\}\text{.}\) Thus the two functions coincide on elements.
Theorem2.8.Generating \(S_n\).
For \(n\ge 2\) prove that \(S_n\) is generated by \((12)\) and the \(n\)-cycle \((12\cdots n)\text{.}\)
Remark2.9.
This theorem will prove surprisingly useful all the way down in Galois Theory, saving us a lot of time with our proof of unsolvable quintic polyomials.
Remark2.10.
Note that in Theorem 1.44 we showed \(S_n\) is generated by transpositions.
Lets look at another example.
Example2.11.Commutator Subgroup.
Let \(G\) be a group, \(S=\{aba\inv b\inv|a,b\in G\}\text{,}\) and \(G'=\igen S\text{.}\) This is known as the commutator subgroup of \(G\text{.}\)
Remark2.12.
Commutator subgroups prove invaluable when it comes to something called abelianization, which is, loosely speaking, a way of "modding out" the non-commutative part of \(G\) to obtain a new group that is abelian. This is seen rigorously in Example 3.43.
A nice property of a Generated Subgroup is that once you’ve located the generators, you’ve found the whole group:
Proposition2.13.Generators of Subgroups.
If \(H\) is a subgroup of \(G\) that contains \(S\text{,}\) then \(\igen S\leq H\text{.}\)
In this way, if we can understand something about the generators of a group, we can (for the most part) extend that knowledge to the group as a whole. This is something you may have seen in the form of a basis, either for a vector space or a topological space.
Summary
For a subset \(S\) of a group, the Generated Subgroup of \(S\) is the smallest subgroup containing \(S\text{.}\)
\(\langle S \rangle = \left\{x_1^{j_1} \cdots x_m^{j_m} \mid m \geq 0, j_1, \dots, j_m \in \mathbb{Z}\text{ and }x_1, \dots, x_m \in S \right\}\text{.}\) 1
