Read: Continue with Section 3.3.
Turn in: 3.51, 3.52, 3.54, 3.57, 3.58, 3.59
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There are multiple ways to approach 3.51 and 3.52. If you want a hint on these, read on; if not, don’t. For 3.51, you could consider the fact that $\phi(e_1) = \phi(e_1*e_1)$, and for 3.52, you could meditate on the equation $\phi(e_1) = \phi(g*g^{-1})$.
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For 3.54, 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.)
Extra practice: 3.47, 3.53, 3.55, 3.56
- For 3.47, 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.
- For 3.53, remember that since $\phi:G_1 \rightarrow G_2$ is a bijection, $\phi^{-1}: G_2\rightarrow G_1$ is a function and is also a bijection. You do not need to prove this. Also, remember the relationship between a function and its inverse: $\phi^{-1}(\phi(x)) = x$ and $\phi(\phi^{-1}(y)) = y$. To prove that $\phi^{-1}: G_2\rightarrow G_1$ has the homomorphic property, you should start by considering $\phi^{-1}(y_1+y_2)$ for $y_1,y_2 \in G_2$. You will need to find a way to rewrite $y_1$ and $y_2$ to move forward.
