Groups HT23, Subgroups
Flashcards
@Define 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 $g _ 1 \ast g _ 2 \in H$.
@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 $G$.