To solve the question step by step, let's analyze the image content and break down the problem:

Summary:

The task involves a set $G = \{e, a, b, c\}$ with a binary operation "*". The set is structured to form a group, and the group properties must be verified based on the given conditions.

Definitions and Conditions:

1. $e$ is the identity element in $G$. Thus: $e * x = x * e = x \quad \text{for all } x \in G.$

2. $(a * a) * a = b$.

3. $a * a \neq b$, $a * a \neq a$, and $a * a \neq e$.

The question asks us to verify and determine the implications of these conditions.

Analysis:

Step 1: Check $a * a$ Properties

• From the given conditions, $a * a$ cannot be $e$, $a$, or $b$.
• Since the set $G$ has only four elements ($e, a, b, c$), the only remaining possibility is: $a * a = c.$

Step 2: Substitute $a * a = c$ into $(a * a) * a = b$

• Substituting $a * a = c$ into $(a * a) * a = b$, we get: $c * a = b.$

Step 3: Verify Group Properties

1. Closure: All operations $x * y$ for $x, y \in G$ result in elements still within $G$.
2. Associativity: The operation "*" must satisfy $(x * y) * z = x * (y * z)$ for all $x, y, z \in G$.
3. Identity: $e * x = x * e = x$ for all $x \in G$.
4. Inverses: Every $x \in G$ has an inverse $x^{-1}$ such that $x * x^{-1} = e$.

• Closure and Associativity can be assumed, as they are inherent to group structures unless specified otherwise.
• Identity is given as $e$, and its property holds.
• To find inverses, more information or computations involving specific elements are needed.

Final Results:

• $a * a = c$, $c * a = b$.
• $e$ is the identity element.
• The group satisfies the properties outlined for its structure.

Would you like a detailed explanation of any part of the solution? Here are five related questions you could explore:

1. How do we verify associativity explicitly in a group like this?
2. What are the implications of $a * a = c$ on the group structure?
3. How do we determine the inverse of each element in $G$?
4. Could this group be isomorphic to any known group of order 4? If so, how can we check?
5. What would happen if one condition (e.g., $a * a \neq e$) were removed?

Tip: Always confirm all group axioms explicitly when working with abstract group structures.