Groups and Group Actions: Lecture 11

Theorem of the week 2018-03-12

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 $|G| = |G/K| \cdot |K|$ .