Groups Sheet 1

Table of Contents

Question 6

(a) We define a homomorphism $\phi_B$: $A \times B \mapsto B$, where $\phi_B(a,b)=b$. This is clearly a surjective homomorphism and it has kernel

\[ker(\phi_B)=\{(a,e_{B}) | a \in A\}=A\]

Now by the first isomorphism thm. $G/ A \cong B$. The other case is analogous.

(b) It’s easy to check that $A_1 \times B_1$ is indeed a normal subgroup of $G$. We construct a homomorphism

\[\phi : G / (A_1 \times B_1) \mapsto A/A_1\times B/B_1\] \[(a,b)A_1 \times B_1 \mapsto aA_1 \times bB_1\]

The multiplicativity and surjective are immediate. For injectivity, we have:

\[\begin{aligned} \phi ((a,b)A_1 \times B_1) &= A_1 \times B_1 \\ aA_1 &=A_1 \\bB_1 &=B_1 \end{aligned}\]

So $(a,b)A_1 \times B_1=A_1 \times B_1$ and $\phi$ is an isomorphism. From this we can induct to get the result in the notes.