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To show that the algebraic structure \((\mathbb{Z}_5, +)\) is a group, we need to verify the four group axioms: closure, associativity, identity, and invertibility. Let's go through each one:

### 1. Closure
The set \(\mathbb{Z}_5\) consists of elements \(\{0, 1, 2, 3, 4\}\). Under the operation of addition modulo 5, adding any two elements \(a, b \in \mathbb{Z}_5\) results in another element \(c \in \mathbb{Z}_5\). Specifically, the sum \(a + b\) is calculated modulo 5, ensuring that the result is always within the set \(\{0, 1, 2, 3, 4\}\).

**Example:**
- \(3 + 4 = 7 \equiv 2 \mod 5\)
- \(1 + 2 = 3 \mod 5\)

Since adding any two elements of \(\mathbb{Z}_5\) results in another element of \(\mathbb{Z}_5\), closure is satisfied.

### 2. Associativity
The addition operation in \(\mathbb{Z}_5\) inherits associativity from the usual addition of integers. That is, for any \(a, b, c \in \mathbb{Z}_5\),
\[
(a + b) + c = a + (b + c)
\]
holds true, because the addition of integers is associative.

### 3. Identity Element
The identity element in a group under addition is the element \(e\) such that for any element \(a \in \mathbb{Z}_5\),
\[
a + e = e + a = a
\]
In \(\mathbb{Z}_5\), the identity element is \(0\), since \(a + 0 = 0 + a = a\) for any \(a \in \mathbb{Z}_5\).

### 4. Inverses
For each element \(a \in \mathbb{Z}_5\), there must exist an element \(b \in \mathbb{Z}_5\) such that
\[
a + b \equiv 0 \mod 5
\]
Each element in \(\mathbb{Z}_5\) has an inverse as follows:
- The inverse of \(0\) is \(0\).
- The inverse of \(1\) is \(4\) because \(1 + 4 = 5 \equiv 0 \mod 5\).
- The inverse of \(2\) is \(3\) because \(2 + 3 = 5 \equiv 0 \mod 5\).
- The inverse of \(3\) is \(2\) because \(3 + 2 = 5 \equiv 0 \mod 5\).
- The inverse of \(4\) is \(1\) because \(4 + 1 = 5 \equiv 0 \mod 5\).

Since all the group axioms (closure, associativity, identity, and inverses) are satisfied, \((\mathbb{Z}_5, +)\) is indeed a group.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What is a cyclic group, and is \((\mathbb{Z}_5, +)\) a cyclic group?
2. How would you show that \((\mathbb{Z}_5, +)\) is an abelian group?
3. Can you describe the subgroup structure of \((\mathbb{Z}_5, +)\)?
4. What is the significance of the order of an element in a group?
5. How does the concept of modular arithmetic apply to cryptography?

**Tip:** In general, for a set with a binary operation to be a group, always check the four fundamental properties: closure, associativity, identity, and invertibility.

Given the algebraic structure, (Z_5, +), a. Show that it is a group.

Explore the proof that the algebraic structure (Z_5, +) forms a group under addition modulo 5. This solution includes verification of group axioms such as closure, associativity, identity, and invertibility. Ideal for high school and university students studying abstract algebra and group theory.

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