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Week 5 Problems

These problems come from the Chapter 6 exercises starting on page 78 of the text.

  1. (Problem 3) Prove that every subgroup of the integers has finite index.

  2. (Problem 5) List the left and right cosets of the subgroups:

    • $\langle 8\rangle$ in $\mathbb{Z}_{24}$.
    • $\mathbb{T}\subset\mathbb{C}^{\times}$. (Recall that $\mathbb{T}$ are the complex numbers of norm one, and $\mathbb{C}^{\times}$ is the multiplicative group of nonzero complex numbers.)
    • $SL_{2}(\mathbb{R})\subset GL_{2}(\mathbb{R})$. Recall that the first group is the set of $2\times 2$ real matrices with determinant $1$, and the second is the set of $2\times 2$ matrices with nonzero determinant.
  3. (Problem 11) Prove Lemma 6.3 in the text.

  4. (Problem 14) Suppose that $g\in G$ and $g^{n}=e$. Prove that the order of the element $g$ divides $n$.

  5. (Problem 16) Suppose that $G$ is a group with an even number of elements. Prove that the number of elements of order $2$ is odd. Conclude that $G$ must contain a subgroup of order $2$. Hint: zero is even.

  6. (Problem 18) If $H$ is a subgroup of $G$ of index two, prove that $gH=Hg$ for all $g\in G$.

  7. (Problem 12) Suppose that $H$ has the property that, for all $g\in G$, we have $gHg^{-1}=H$ where \(gHg^{-1}=\{ghg^{-1}: h\in H\}\) Prove that $gH=Hg$ for all $g$, so the left and right cosets of $H$ are the same.

  8. (Problem 15) The cycle structure of a permutation $\sigma$ is the unordered list of the sizes ofthe cycles in its cycle decomposition. So, for example, the permutation \(\sigma = (12)(345)(78)(9)\) has cycle structure $(2,3,2,1)$ or, equivalently, $(1,2,2,3)$. Prove that two permutations $\alpha,\beta\in S_{n}$ have the same cycle structure if and only if there exists a permutation $k$ such that $\beta = k\alpha k^{-1}$. If this holds, we say the permutations are conjugate.