Let $G$ be a group, $H$ be a subgroup of $G$, and let $a, b \in G$. We want to prove that the left coset $aH$ and the right coset $Hb$ of $H$ in $G$ have the same cardinality, i.e., $|aH| = |Hb|$.

Step-by-Step Proof:

1. Define the cosets:

• The left coset of $H$ with respect to $a$ is given by: $aH = \{ ah \mid h \in H \}.$
• The right coset of $H$ with respect to $b$ is given by: $Hb = \{ hb \mid h \in H \}.$

We need to show that these two sets, $aH$ and $Hb$, have the same size (i.e., the same number of distinct elements).

2. Construct a bijection between $aH$ and $Hb$:

Consider the map $f : aH \to Hb$ defined by: $f(ah) = hb \quad \text{for all } h \in H.$ This map takes each element of the left coset $aH$, which is of the form $ah$, and maps it to an element of the right coset $Hb$, namely $hb$.

3. Show that $f$ is well-defined:

For $f(ah) = hb$ to be well-defined, we must check that if two elements in $aH$ are equal, their images under $f$ must also be equal.

Suppose $ah_1 = ah_2$ for some $h_1, h_2 \in H$. Then, by left-multiplying both sides by $a^{-1}$, we get: $h_1 = h_2.$ Therefore, $f(ah_1) = f(ah_2)$, meaning the map is well-defined.

4. Show that $f$ is injective (one-to-one):

Suppose $f(ah_1) = f(ah_2)$, meaning: $hb_1 = hb_2 \quad \text{for some } h_1, h_2 \in H.$ This implies that: $h_1 = h_2.$ Therefore, $f$ is injective.

5. Show that $f$ is surjective (onto):

For any element $x = hb \in Hb$, there exists a corresponding element $x = ah \in aH$ under the map $f$. Therefore, the map is surjective.

Since $f$ is both injective and surjective, it is a bijection between the sets $aH$ and $Hb$. Therefore, the two sets must have the same cardinality.

Conclusion:

Since $f$ is a bijection, we conclude that: $|aH| = |Hb|.$

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Additional Exploration:

Would you like further details on any specific part of the proof? Here are some related questions that might help deepen your understanding of this topic:

1. How do cosets partition a group into disjoint subsets?
2. Can we apply a similar argument to prove that left cosets and right cosets of a normal subgroup have the same size?
3. How does the index of a subgroup relate to the number of cosets?
4. How can we prove that the left and right cosets of any subgroup $H$ in $G$ have the same cardinality using Lagrange’s Theorem?
5. What happens if $H$ is a normal subgroup in terms of coset equivalence?

Tip: A good strategy for proving set equalities is to construct a well-defined bijection, which is often the simplest way to show two sets have the same cardinality.