To determine if the system defined in the table forms a group, we need to check the four properties of a group:

1. Closure: For every pair of elements $a$ and $b$ in the set, the result of the operation $a \ast b$ must also be in the set.
2. Associativity: For every $a, b, c$ in the set, $(a \ast b) \ast c = a \ast (b \ast c)$.
3. Identity element: There must exist an element $e$ in the set such that for every element $a$, $e \ast a = a \ast e = a$.
4. Inverses: For each element $a$ in the set, there must exist an element $b$ such that $a \ast b = b \ast a = e$.

Step 1: Define the table and elements

The operation table provided is:

asterisk	0	2	4
0	0	0	0
2	0	4	8
4	0	8	16

Step 2: Check Closure

• The results of the operation for all combinations of the elements (0, 2, 4) are in the set {0, 2, 4, 8, 16}.
• Closure is satisfied.

Step 3: Check Associativity

To verify associativity, we would need to check all combinations, which can be tedious without clear patterns. However, we can test a few combinations:

• For example: $(2 \ast 2) \ast 4 = 4 \ast 4 = 16$ and $2 \ast (2 \ast 4) = 2 \ast 8 = 0$.
• Since $(2 \ast 2) \ast 4 \neq 2 \ast (2 \ast 4)$, associativity is not satisfied.

Step 4: Check Identity

From the table, $0$ appears to be an identity for some elements:

• $0 \ast 0 = 0$, $0 \ast 2 = 0$, $0 \ast 4 = 0$.
• However, $2 \ast 0 \neq 2$ and $4 \ast 0 \neq 4$.
• Thus, there is no identity element.

Step 5: Check Inverses

Since there is no identity element, we cannot find inverses.

Conclusion

The system does not satisfy the associativity property, the identity property, and consequently, the inverse property.

The correct answer is: No, the identity and inverse properties are not satisfied.

Questions for Further Exploration

1. What examples of mathematical systems do satisfy all group properties?
2. Can a set be closed under an operation but still not form a group?
3. How can we demonstrate associativity using specific element combinations?
4. What modifications could be made to this operation to form a group?
5. How does the concept of a monoid differ from a group?

Tip: Always verify the properties of a group systematically to avoid missing critical aspects.