Groups HT23, Normal subgroups
Flashcards
@Define a normal subgroup $H$ of $G$.
$H \trianglelefteq G$ if
\[gH = Hg\]for all $g \in G$, or equivalently
\[g^{-1}hg \in H\]for all $g \in G$, $h \in H$.
@Define a simple group.
A group where $G$ and $\{e _ G\}$ are the only normal subgroups.
Given that $H$ will be a normal subgroup of $G$ if and only if
\[g^{-1}hg \in H\]
for all $g \in G$, $h \in H$, what does this mean in terms of conjugacy classes?
$H$ consists of the union of a number of conjugacy classes.
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?
It is a normal subgroup.
Let $G$ be a group. @Define 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?
normal subgroup
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?
Show it’s the kernel of a homomorphism.
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?
Consider trivial conjugacy classes, i.e. $r^2$.