To solve this problem, let's analyze the given information step by step.

Information Given:

1. $G$ is a group under the operation $*$ with identity $e$.
2. $a, b \in G$ satisfy $a * b = b * a$, meaning $a$ and $b$ commute.
3. $a * a * a * b = e$ (equation 1).
4. $a * a * b * b = e$ (equation 2).

We need to find $a * b$.

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Solution:

Step 1: Use Equation 1

From $a * a * a * b = e$, we can write: $a^3 * b = e$ Since $b$ commutes with $a$, this implies: $b = a^{-3}.$

Step 2: Substitute $b = a^{-3}$ into Equation 2

Using $a * a * b * b = e$, substitute $b = a^{-3}$: $a * a * a^{-3} * a^{-3} = e.$ Simplify $a * a = a^2$ and $a^{-3} * a^{-3} = a^{-6}$: $a^2 * a^{-6} = e.$ Combine exponents using the group property: $a^{2-6} = e.$ Thus: $a^{-4} = e.$

Step 3: Simplify $a^{-4} = e$

The equation $a^{-4} = e$ implies $a^4 = e$. So, the order of $a$ divides 4.

Step 4: Find $a * b$

Using $b = a^{-3}$, calculate $a * b$: $a * b = a * a^{-3}.$ Simplify: $a * b = a^{1-3} = a^{-2}.$

Since $a^4 = e$, we know $a^{-2} = a^2$. Therefore: $a * b = a^2.$

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Final Answer:

$a * b = a^2.$

Let me know if you have questions or need further details!

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Related Questions:

1. What is the significance of commutativity in simplifying group operations?
2. How does $a^4 = e$ affect the properties of the group $G$?
3. If $a * b = b * a$, can this result generalize to all elements of $G$?
4. What is the impact of different orders of $a$ on the solution?
5. How would the solution change if $a^4 \neq e$?

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Tip:

In group theory, always verify the group's properties (associativity, identity, inverses) and use commutativity to simplify expressions when given.