# Notes - Groups HT23, Subgroups

### Flashcards

What is the formal definition of a subgroup $H \le G$?

Let $G$ be a group. A subset $H \subseteq G$ is a subgroup of $G$ if

- $e \in H$
- If $g \in H$, then $g^{-1} \in H$.
- If $g _ 1, g _ 2 \in H$, then $g _ 1 \ast g _ 2 \in H$.

Can you state the subgroup test?

Let $G$ be a group. A subset $H \subseteq G$ is a subgroup of $G$ if and only if

\[\forall g_1, g_2 \in H \implies g_1^{-1}g_2 \in H\]What is true about the intersection of any number of subgroups of some group $G$?

It is also a group.

Let $G$ be a group and $S$ a subset of $G$. What does the notation $\langle S \rangle$ mean?

The smallest subgroup containing all elements of $S$.

How could you write $\langle S \rangle$, the subgroup generated by $S$, as an intersection?

“the intersection of all subgroups of $G$ which contain $S$”.

What do you call the elements of $S$ if $G = \langle S \rangle$?

Generators of $S$.

Can you give a set that generates $D _ {2n}$?

Given that every permutation can be written as a product of transpositions, can you give a set that generates $S _ n$?

### Proofs

Prove that if $\{H _ i\} _ {i\in I}$ is a collection of subgroups of a group $G$ then

\[\bigcap _ {i \in I} H _ i\]
is a subgroup of $G$.

Todo.