# Notes - Groups HT23, Cosets

### Flashcards

Let $H$ be a subgroup of $G$. What are the left cosets of $H$?

\[gH = \\{gh : h \in H\\}\]

Let $H$ be a subgroup of $G$. What are the right cosets of $H$?

\[Hg = \\{hg : h \in H\\}\]

What notation is used for the left cosets of $H$ in $G$, other than $gH$?

\[G/H\]

Let $H$ be a subgroup of $G$. Define the index of $H$ in $G$.

\[|G/H|\]

Can you state the coset equality lemma?

Let $H \leqslant G$ and $g, k \in G$. Then

\[gH = kH \iff k^{-1}g \in H\]and

\[Hg = Hk \iff kg^{-1} \in H\]If $g, k \in G$, what equivalence relation $g \sim k$ means that the equivalence classes are the left cosets of $H$?

\[g\sim k \iff k^{-1}g \in H\]