#8 please Chapter 3 Homework1. Prove that in any group, an element and its inverse have the same order.2. Let \( x \) be in a group \( G \). If \( x^* = e \) and \( x^{**} = e \), prove that \( x^{**} = e \) and \( x^{**} = e \).3. Show that \( Z_{10} = x^3 \equiv 3 \, (\text{mod} \, 10) \).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: | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |---|---|---|---|---|---|---|---|---| | 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | 2 | 2 | 1 | 4 | 3 | 6 | 5 | 8 | 7 | | 3 | 3 | 4 | 1 | 2 | 7 | 8 | 5 | 6 | | 4 | 4 | 3 | 2 | 1 | 8 | 7 | 6 | 5 | | 5 | 5 | 6 | 7 | 8 | 1 | 2 | 3 | 4 | | 6 | 6 | 5 | 8 | 7 | 2 | 1 | 4 | 3 | | 7 | 7 | 8 | 5 | 6

Name: #8 please Chapter 3 Homework1. Prove that in any group, an element and
Uploaded: 2024-03-20T07:06:41.000Z
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Description: #8 please Chapter 3 Homework1. Prove that in any group, an element and its inverse have the same order.2. Let \( x \) be in a group \( G \). If \( x^* = e \) and \( x^{**} = e \), prove that \( x^{**} = e \) and \( x^{**} = e \).3. Show that \( Z_{10