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Week 2 Problems

These problems are drawn from Chapter 3 of the textbook. They are based on the material from Week 1 and will be discussed in class during Week 2.

  1. (Problem 3) Let $G$ be the group of symmetries of a non-square rectangle, and let $H$ be the group $\mathbb{Z}_{4}$ with addition as the operation. How many elements are in eqch group? Are they the same? Why or why not? Hint: look at Cayley tables.

  2. (Problem 6) Give a multiplication table for the group $U(12)$.

  3. (Problem 8) Prove that $\mathrm{GL}_{2}(\mathbb{R})$ is not an abelian group.

  4. (Problem 10) Let $H$ be the set of upper triangular, $3x3$ matrices with ones on the diagonal and real entries above the diagonal. Prove that $H$ is a group under matrix multiplication. This group is called the Heisenberg Group and it is important in quantum mechanices.

  5. (Problem 16) Given an example of a group $G$ with two elements $g,h$ such that $(gh)^2\not=g^2h^2$.

  6. (Problem 45) Prove that if $H$ and $K$ are subgroups of a group $G$, then so is $H\cap K$. What about $H\cup K$?

  7. (Problem 32) Suppose $G$ is a finite group with an even number of elements. Prove that there is an $a$ in $G$ such that $a\not=e$ and $a^2=e$.

  8. (Problem 48) Let $G$ be a group. Define a subset $Z(G)$ by

\[Z(G) = \{x\in G: gx=xg \mathrm{\ for\ all\ }g\in G\}\]

Prove that $Z(G)$ is a subgroup of $G$. $Z(G)$ is called the center of $G$.