Read: Section 3.3.
Turn in: 3.51, 3.54, 3.56, 3.61, 3.64, 3.65
- For 3.51, you are given a function and need to check three things: (1) does it satisfy the homomorphic property in Definition 3.48, (2) is it one-to-one, and (3) is it onto. If any one of them fails, it is not an isomorphism.
- There are multiple ways to approach 3.54. If you want a hint, read on: one approach is to consider the equation $\phi(e_1) = \phi(g*g^{-1})$.
- For 3.56 you can use (without proof) that the composition of bijections is a bijection. (A function is a bijection if it is one-to-one and onto.)
- For 3.64 you will need to make use of the fact that $\phi$ is onto; this implies that every element $y \in G_2$ can be written as $y = \phi(x)$ for some $x\in G_1$. Carefully write down what you want to prove about $G_2$, and then use use that $\phi$ is onto to translate what you want to prove.
- The hardest part of 3.65 is to make sense of the definition $\phi(H)$. When you get to proving that $\phi(H)$ is a subgroup of $G_2$, you should find that you only need to use that $\phi$ satisfies the homomorphic property; it won’t be necessary that $\phi$ is one-to-one and onto.
Extra practice: 3.49, 3.55, 3.57, 3.58, 3.59, 3.67
- For 3.49, your answers should look like lists of isomorphic groups with a short justification like “$D_4 \cong \operatorname{Spin}_{1\times 2}$ as shown by the matching we found in 3.35.” As you explore some of the groups that we hadn’t thought of before, here are some things to keep in mind: (1) isomorphisms are bijections so groups of different sizes cannot be isomorphic, and (2) isomorphic groups must share the same group-theoretic properties so if one group has 6 elements with the property that $g^2=e$ and another group only has 2 elements with that property, then they can’t be isomorphic.
