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The image shows three group theory problems. I'll transcribe them below:

1. (Not fully visible in the image)
2. **Let \( G \) be a group and \( c, d \in G \). Show that the equation \( xc = d \) has a unique solution in \( G \).**
3. **Let \( a \) and \( b \) be two elements of a group. If the order of \( a \) is \( d \), then show that the order of \( a^{-1} \) is also \( d \), and that \( o(bab^{-1}) = o(a) \), where \( o(a) \) denotes the order of \( a \).**

Would you like a step-by-step solution for one or both of these problems?

Here are some related questions:

1. What is the definition of the order of an element in a group?
2. How can we prove that every element in a group has an inverse?
3. Why does conjugation preserve the order of elements in a group?
4. What properties must a group satisfy to ensure the uniqueness of solutions?
5. How can the associative property be used in the context of these problems?

**Tip:** When working with group theory, always keep the basic group properties (associativity, identity, and inverses) in mind—they are key to solving most problems.

Let G be a group and c, d ∈ G. Show that the equation xc = d has a unique solution in G.
Let a and b be two elements of a group. If the order of a is d, then show that the order of a^-1 is also d, and that o(bab^-1) = o(a).

This page provides step-by-step solutions to two group theory problems: showing the uniqueness of solutions in groups and proving properties of element orders and conjugation in groups. Suitable for undergraduate students studying abstract algebra.

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