Groups TT23, Quotient groups
Flashcards
If $H \trianglelefteq G$ then what is the quotient group?
\[(G/H, \ast)\]
where $\ast$ is given by
\[(g _ 1 H) \ast (g _ 2 H) = (g _ 1 g _ 2) H\]for $g _ 1 H, g _ 2 H \in G/H$.
What is some alternative notation for $\mathbb Z / n\mathbb Z$?
\[\mathbb Z _ n, C _ n\]
Why does $G/H$ correspond to “modding out” $H$ when $H$ is a normal subgroup?
$H$ can be thought of a set of things that we want to form an equivalence class, and then each $gH$ can be thought of like a translated version of that equivalence class.