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 $G$.
\[ \vert G/H \vert \]
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 $H$?
\[g\sim k \iff k^{-1}g \in H\]