Normal Subgroups: Conjugation, Quotients, and Homomorphisms

All flashcards

What is a normal subgroup?

A subgroup $N$ of $G$ such that $gNg^{-1}=N$ for every $g\in G$.

What is the kernel of a group homomorphism $f:G\to H$?

The subgroup $\{g\in G:f(g)=e_H\}$, consisting of elements mapped to the identity of $H$.

What is the normal core of a subgroup?

The largest normal subgroup of $G$ contained in that subgroup.

What is a Hamiltonian group?

A nonabelian group in which every subgroup is normal.

Which subgroup is normal in a semidirect product $G=N\rtimes H$?

The subgroup $N$ is normal in $G$; the subgroup $H$ need not be.

Quiz questions

1. For a group homomorphism $f:G\to H$, where is $\ker f$ necessarily normal?Answer: In the domain group $G$.
2. Which general statement holds for subgroups of index two?Answer: Every subgroup of index two is normal.
3. What does the first isomorphism theorem state for a homomorphism $f:G\to H$?Answer: $G/\ker f\cong f(G)$.
4. If $N\triangleleft G$ and $N\leq K\leq G$, what follows?Answer: $N\triangleleft K$.
5. What can be concluded about subgroups of an abelian group?Answer: Every subgroup is normal.