Groups and Group Actions Lecture 10

In which we think some more about homomorphisms, and meet normal subgroups.

• Proposition 45: Let $\theta : \mathbb{Z} \to \mathbb{Z}$ be a homomorphism.  Then there is $n \in \mathbb{Z}$ with $\theta(m) = nm$ for all $m \in \mathbb{Z}$.  We proved this by defining $n = \theta(1)$ and then using the fact that $1$ generates $\mathbb{Z}$.
• Definition of the kernel and image of a homomorphism.
• Proposition 46: Let $G$, $H$ be groups, let $\theta : G \to H$ be a homomorphism.  Then
1. $\ker \theta$ is a subgroup of $G$; and
2. $\mathrm{Im} \theta$ is a subgroup of $H$.

We proved this using the subgroup test.

• Proposition 47: Let $G$, $H$  be groups, let $\theta: G \to H$ be a homomorphism.  Then $\theta$ is constant on each coset of $\ker \theta$, and takes different values on different cosets.  We saw that $\theta(g_1) = \theta(g_2)$ if and only if $g_1 g_2^{-1} \in \ker\theta$, and then used Lemma 33 to see that this is equivalent to $g_1 \ker \theta = g_2 \ker \theta$.
• Corollary 48: Let $G$, $H$ be groups, let $\theta: G \to H$ be a homomorphism.  Then $\theta$ is injective if and only if $\ker \theta = \{ e_G \}$.  This was immediate from Proposition 47, since $e_G \in \ker \theta$.
• Definition of a normal subgroup of a group.
• Proposition 49: Let $G$, $H$ be groups, let $\theta : G \to H$ be a homomorphism.  Then $\ker\theta$ is a normal subgroup of $G$.  We already know from Proposition 46 that $\ker\theta$ is a subgroup of $G$, so we just checked that if $k \in \ker \theta$ and $g \in G$ then $g^{-1} k g \in \ker\theta$.
• Proposition 50: Let $H$ be a subgroup of a group $G$ with index $|G/H| = 2$.  Then $H \trianglelefteq G$.  We argued that the only (left and right) cosets of $H$ in $G$ are $H$ and $G \setminus H$, and used this to see that $gH = Hg$ for all $g \in G$.

Understanding today’s lecture

Pick some homomorphisms.  Can you identify their kernels and images?  Which homomorphisms are injective?  Which are surjective?

Pick a homomorphism $\theta$ between two groups (pick explicit groups and an explicit homomorphism).  What are the cosets of $\ker \theta$?  This might help you to get a feel for Proposition 47.

Pick some subgroups of groups.  Which are normal?

Of course Wikipedia has a page about normal subgroups, and another about the notion of a simple group.  You will learn more about simple groups if you choose to study Algebra 3 (Group Theory) next year.

MacTutor suggests that Galois was the first to recognise the significance of normal subgroups.

Preparation for Lecture 11

Can you show that if $H$ is a normal subgroup of $G$ then $G/H$ (the set of left cosets of $H$ in $G$) forms a group under the natural operation?

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