Cosets, Normal Subgroups, and Quotient Groups

All flashcards

Normal subgroup

A subgroup $N$ of $G$ is normal if $g^{-1}ng\in N$ for every $g\in G$ and $n\in N$.

Quotient-group operation

For normal $N$, cosets multiply by $(aN)(bN)=(ab)N$.

Canonical projection

The map $\pi:G\to G/N$ defined by $\pi(g)=gN$ is a surjective homomorphism with kernel $N$.

First isomorphism theorem

For a group homomorphism $\varphi:G\to H$, $G/\ker(\varphi)\cong\operatorname{im}(\varphi)$.

Finite quotient order

For finite $G$ and normal $N$, $|G/N|=|G|/|N|$.

Quiz questions

1. Which condition characterizes a normal subgroup $N$ of a group $G$?Answer: $g^{-1}ng\in N$ for every $g\in G$ and $n\in N$.
2. For normal $N$ in $G$, how is the product of cosets $aN$ and $bN$ defined?Answer: $(aN)(bN)=(ab)N$.
3. What does the first isomorphism theorem state for a homomorphism $\varphi:G\to H$?Answer: $G/\ker(\varphi)\cong\operatorname{im}(\varphi)$.
4. If $G$ is finite and $N$ is a normal subgroup, what is the order of $G/N$?Answer: $|G|/|N|$.
5. What condition on a subgroup $N$ of a Lie group $G$ ensures that $G/N$ is a Lie group?Answer: $N$ is closed and normal in $G$.