A cyclic group is generated by just one element. The notion of a “generated subgroup” generalizes this concept to any arbitrary number of generators.

Note that in the definition below, we use the idea of the “smallest” subgroup of G, satisfying a certain condition. What does “smallest” mean?

The smallest subgroup satisfying a condition, is (1) a subgroup which satisfies the condition, and (2) a subset of all other subgroups which also satisfy the condition.

Definition:

Let G be a group and $A\subseteq G$ a nonempty subset.

We define $\langle A\rangle$, the group generated by A, to be the smallest subgroup of G that contains A.

Theorem:

$$ \langle A\rangle = \bigcap_{\substack{H\le G\\A\subseteq H}} H $$

Proof:

The right-hand side is a subgroup, as a consequence of a previous theorem.

It also contains A as a subset.

Exercise:

Prove that $\displaystyle A\subseteq \bigcap_{\substack{H\le G\\A\subseteq H}}H$.

And if $K\le G$ and $A\subseteq K$, then $\displaystyle \bigcap_{\substack{H\le G\\ A\subseteq H}}H\le K$.

Exercise:

Prove the claim above.

By all of the above, we see that $\displaystyle \bigcap_{\substack{H\le G\\ A\subseteq H}}H$ has the property of being $\langle A\rangle$.

Moreover, $\langle A\rangle$ must be unique. This is because for any two subgroups $B,C\le G$ which satisfy the property, we have that B is a subgroup containing A which is a subset of all other subgroups containing A. But the same is true for C.

Therefore $B\subseteq C$ and $C\subseteq B$ and therefore $B=C$.

$\Box$

Exercise:

Show that a rotation and reflection generate $D_{2n}$. That is to say, $\langle r,s\rangle = D_{2n}$.

Find a smallest set of generators of Q, the group of quaternions.