## Groups and Group Actions: Lecture 11

In which we meet and explore quotient groups.

• Definition of the centre of a group.
• Proposition 51: Let $G$ be a group.  Then $Z(G) \trianglelefteq G$.  The proof is an exercise.
• Proposition 52: Let $G$ be a group, let $H$ be a subgroup of $G$.
1. Define a binary operation $\ast$ on $G/H$ via $(g_1 H) \ast (g_2 H) = (g_1 g_2) H$.  This is well defined if and only if$H \trianglelefteq G$.
2. If $H \trianglelefteq G$, then $(G/H, \ast)$ is a group.

Each part was a careful check, relying on the definition of a normal subgroup.

• Definition of a quotient group.
• Proposition 53: Let $G$ be a group, let $H$ be a subset of $G$.  Then $H$ is a normal subgroup of $G$ if and only if it is the kernel of a homomorphism with domain $G$.  One direction was Proposition 49.  For the other, where $H \trianglelefteq G$ so $G/H$ is a group, we defined the quotient map $\pi: G \to G/H$ by $\pi(g) = gH$, and showed that this is a homomorphism with kernel $H$.
• Theorem 54 (First isomorphism theorem): Let $G$, $H$ be groups, let $\theta : G \to H$ be a homomorphism.  Then $G/\ker \theta \cong \mathrm{Im} \theta$, and the map $\tilde{\theta} : G/\ker\theta \to \mathrm{Im} \theta$ given by $\tilde{\theta}(g \ker\theta) = \theta(g)$ is an isomorphism.  We checked that $\tilde{\theta}$ is well defined (using Proposition 47), that it is a homomorphism, that it is injective, and that it is surjective.
• Corollary 55: Let $G$ be a finite group, let $H$ be a group, let $\theta : G \to H$ be a homomorphism.  Then $|G| = |\ker\theta| \cdot | \mathrm{Im} \theta|$.  We used our proof of Lagrange’s theorem, which showed that if $K \leq G$then $|G| = |G/K| \cdot |K|$.

### Understanding today’s lecture

Can you prove Proposition 51 (that the centre of a group is a normal subgroup of that group)?

Can you fill in all the details in the proof of Proposition 52(ii) (that $(G/H, \ast)$ is a group)?

You could explore Corollary 55 by picking some explicit examples and seeing how it works for them.

Tim Gowers has written a really interesting blog post about normal subgroups and quotient groups.  He’s also written about what it means to say that a function is well defined, which was very relevant for today’s lecture.

A long time ago, I wrote something about the first isomorphism theorem.

If, like me, you have recently found yourself trying to remember/discover how to write $\trianglelefteq$ or $\cong$ in LaTeX, then you will appreciate Detexify.

### Preparation for Lecture 12

How many homomorphisms are there from $S_3$ to $C_4 \times C_2$?  (We’ll do this in the lecture, but do have a go before then.)

We’re also going to meet the notion of a group action.  You could read this post by Tim Gowers to start to get a feel for what that’s about.