# Notes - 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 $\mathbb Z / n\mathbb Z$ isomorphic to?

\[\mathbb Z_n\]

What is $S _ n / A _ n$ isomorphic to?

\[C_2\]

What is $\mathbb C^\ast / S^1$ isomorphic to?

\[\mathbb R^\ast\]

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.

### Proofs

Prove that if $H \leqslant G$ then the quotient group defined by

\[(G/H, \ast)\]
and where $\ast$ is given by

\[(g_1 H) \ast (g_2 H) = (g_1 g_2) H\]
is defined if and only if $H \trianglelefteq G$.

Todo (groups and group actions, page 57)