#10 **Chapter 3 Homework**1. Prove that in any group, an element and its inverse have the same order.2. Let \( x \) be in a group \( G \). If \( x^* \ast e \) and \( x^* \ast e^* \), prove that \( x^* \ast e \) and \( x^* \ast e^* \).3. Show that \( Z_{10} = \langle 3 (\text{mod } 10) \rangle \).4. If \( H \) and \( K \) are subgroups of a group \( G \), show that \( HCK \) is a subgroup of \( G \).5. If \( H_{\alpha} : \alpha \in A \) are a family of subgroups of the group \( G \), show that \( \bigcap H_{\alpha} \) is a subgroup of \( G \).6. If \( G \) is a group and \( a \) is an element of \( G \), show that \( C(a) = C(a^{-1}) \).7. Use the following Cayley table for the Group \( G \) to answer 7A, 7B, and 7C. **Cayley Table:** \[ \begin{array}{c|cccccc} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 2 & 3 & 4 & 5 & 6 & 1 \\ 3 & 3 & 4 & 5 & 6 & 1 & 2 \\ 4 & 4 & 5 & 6 & 1 & 2 & 3 \\ 5 & 5 & 6 & 1 & 2 & 3 & 4 \\ 6 & 6 & 1 & 2 & 3 & 4 & 5 \\ \end{array} \] 7A. Given the group \( G \) above, find \( C(a) \) for all \( a \) in \( G \). 7B. Find the

Name: #10 **Chapter 3 Homework**1. Prove that in any group, an element and i
Uploaded: 2024-03-05T11:52:21.000Z
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Description: #10 **Chapter 3 Homework**1. Prove that in any group, an element and its inverse have the same order.2. Let \( x \) be in a group \( G \). If \( x^* \ast e \) and \( x^* \ast e^* \), prove that \( x^* \ast e \) and \( x^* \ast e^* \).3. Show that \(