To show that $G / C$ is an abelian group, we need to prove that for any two elements $g_1C$ and $g_2C$ in the quotient group $G / C$, their product satisfies the commutative property:

$g_1C \cdot g_2C = g_2C \cdot g_1C$

Step-by-step Proof:

1. Recall the definition of $G / C$:

   • $C$ is the commutator subgroup of $G$. It is the smallest normal subgroup of $G$ that contains all commutators of $G$. A commutator is an element of the form $aba^{-1}b^{-1}$ for some $a, b \in G$.
   • The quotient group $G / C$ consists of cosets of the form $gC$, where $g \in G$.

2. Commutativity in $G / C$:

   • We want to show that for any $g_1, g_2 \in G$, the cosets $g_1C$ and $g_2C$ commute, i.e., $g_1C \cdot g_2C = g_2C \cdot g_1C$.

3. Expand both products: $g_1C \cdot g_2C = (g_1g_2)C$ and $g_2C \cdot g_1C = (g_2g_1)C.$

4. Relating these two products:

   • The products $g_1g_2$ and $g_2g_1$ differ by the commutator of $g_1$ and $g_2$: $g_1g_2 = g_2g_1 \cdot [g_1, g_2],$ where $[g_1, g_2] = g_1g_2g_1^{-1}g_2^{-1}$ is the commutator of $g_1$ and $g_2$.

5. Use the fact that $C$ contains all commutators:

   • Since $C$ is the commutator subgroup, the commutator $[g_1, g_2] \in C$. Therefore, we can write: $g_1g_2 = g_2g_1 \cdot c$ for some $c \in C$.

6. In the quotient group $G / C$:

   • In the quotient group, elements of $C$ are equivalent to the identity element. Therefore, we have: $(g_1g_2)C = (g_2g_1c)C = (g_2g_1)C,$ because $c \in C$, so multiplying by $c$ does not change the coset.

7. Conclusion:

   • Thus, $g_1C \cdot g_2C = g_2C \cdot g_1C$, which means that $G / C$ is an abelian group.

Final Statement:

Since $G / C$ satisfies the commutative property, we conclude that $G / C$ is an abelian group.

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Would you like a more detailed explanation of any specific part of this proof?

Here are 5 related questions to expand on this topic:

1. What is the significance of a commutator in group theory?
2. Why is the commutator subgroup the smallest normal subgroup containing all commutators?
3. How can we interpret the structure of the quotient group $G / C$ geometrically?
4. How does the commutator subgroup relate to solvable groups?
5. Can we provide an example of a group $G$ where the commutator subgroup is non-trivial?

Tip: A group is abelian if and only if its commutator subgroup is trivial.