# Largest Normal subgroup

let $G$ be a finite group and $H$ any subgroup. $\psi_{H}$ be the left action of $G$ on $G/H$. It was asked to prove that the action is transitive and the kernel of $\psi_{H}$ is the 'largest normal subgroup'. It was easy to see that it is transitive.

What does this 'largest normal subgroup' mean? Is it some thing anologous to maximal ideal definition? Or does it mean the normal subgroup with greatest cardinality? (note that $G$ need not be finite). Thanks

Edit: By $G/H$ I mean the set of left cosets.

• I'm fairly sure that your source meant: "Largest normal subgroup $N$ of $G$ contained in $H$". And largest means with respect to inclusion: if $K\unlhd G$ and $K\subseteq H$, then $K\subseteq N$. – Jyrki Lahtonen Sep 13 '11 at 10:10
• @Jyrki Actually I was asked this problem by a stranger many days ago. I just noted down that in a hurry. So it is possible that what you said is fairly correct :-) – Dinesh Sep 13 '11 at 10:12
• @Dinesh: This fact is proven in $\text{Herstein's}$ Topics in Algebra book. See page 73 which has the following theorem: If $G$ is a group $H \leq G$ and $S$ is the set of all right cosets of $H$ in $G$, then there is a homomorphism $\theta : G \to \mathscr{A}(S)$ and the kernel of $\theta$ is the $\text{largest normal subgroup}$ of $G$ which is contained in $H$. – user9413 Nov 1 '11 at 20:08
• @Chandrasekhar thanks! – Dinesh Nov 1 '11 at 20:11
• @Dinesh: You are always welcome :) – user9413 Nov 1 '11 at 20:12

It should be the largest normal subgroup of $G$ contained in $H$. The thing about normal subgroups $N_1, N_2$ is that $N_1N_2$ is still a normal subgroup. That means that the normal subgroup contained in $H$ of greatest cardinality is unique (if there are two, take their product and get a bigger one).
This object is also called the core of $H$, and equal to the intersection of all conjugates of $H$ in $G$, i.e. $\bigcap _{g \in G} g^{-1}Hg$. See http://en.wikipedia.org/wiki/Core_%28group%29
For infinite groups, measuring "largest" by cardinality isn't necessarily the right thing to do. You can look at the poset (in fact, lattice) of normal subgroups. The core is still the unique maximal (by inclusion) normal subgroup contained in $H$.