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To prove that the \( H \)-equivalence classes of a group \( G \) are precisely the left cosets of \( H \) in \( G \), let's proceed by understanding the equivalence relation and showing how it relates to the cosets.

### Proof Outline

1. **Define the Equivalence Relation**:
   Define an equivalence relation \( \sim \) on \( G \) such that \( a \sim b \) if and only if \( a^{-1}b \in H \), where \( H \) is a subgroup of \( G \).

2. **Show That Equivalence Classes are Left Cosets**:
   - If \( a \sim b \), then \( a^{-1}b \in H \).
   - Rearranging, we get \( b = a h \) for some \( h \in H \), which implies that \( b \in aH \), the left coset of \( H \) containing \( a \).
   - Therefore, every element equivalent to \( a \) is in the left coset \( aH \), so the equivalence class of \( a \) is exactly \( aH \).
   
3. **Conclude the Equivalence**:
   Thus, each equivalence class under \( \sim \) corresponds to a unique left coset of \( H \) in \( G \). Consequently, the \( H \)-equivalence classes are precisely the left cosets of \( H \).

Would you like further details on any part of this proof or additional questions related to cosets and equivalence relations?

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### Related Questions
1. What is the definition of a left coset and how does it differ from a right coset?
2. How do we know that the relation defined is an equivalence relation?
3. What would change if we considered right cosets instead of left cosets?
4. Can you explain the significance of cosets in group theory and their applications?
5. How would this proof change if \( H \) were not a subgroup of \( G \)?

### Tip
Remember, equivalence relations on groups often help classify elements into distinct subsets (like cosets), which is a useful way to study group structure.

Prove that the H-equivalence classes of G are precisely the left cosets.

This page provides a detailed proof that the H-equivalence classes of a group G are precisely the left cosets. It explores group theory concepts, including equivalence relations and cosets, making it ideal for college-level mathematics students.

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