Notes - Groups HT23, Normal subgroups

What is the definition of a normal subgroup $H$ of $G$?

What is a simple group?

Given that $H$ will be a normal subgroup of $G$ if and only if

for all $g \in G$, $h \in H$, what does this mean in terms of conjugacy classes?

Can you give an example of a non-Abelian simple group?

Let $H \leqslant G$. What fact about the index of $H$ in $G$ allows you to conclude $H$ is a normal subgroup?

What is special about the subgroup formed by the kernel of any homomorphism?

Let $G$ be a group. What is the centre of $G$, denoted $Z(G)$?

What is always true about the relationship between centre of a group $Z(G)$ and the group itself?

Other than showing that $gH = Hg$ or that $g^{-1}hg \in H$ for all $g$, what’s an alternative way that you could show $H$ is a normal subgroup of some group?

When asked to find the centre of a group, e.g. of $D _ 8$, what can you use to do this rather than considering everything invariant under conjugation?

Prove the equivalence in definitions for a normal subgroup $H$ or $G$, that $H \trianglelefteq G$ if

for all $g \in G$, or equivalently

for all $g \in G$, $h \in H$.

Prove that if $ \vert G/H \vert = 2$ then $H \leqslant G$ is a normal subgroup.

Prove that the kernel of any homomorphism $\phi : G \to H$ is a normal subgroup of $G$.