The group $\mathbb{Z}_5$ has an order of 5.
Understanding the Group $\mathbb{Z}_5$
The notation $\mathbb{Z}_5$ (often read as "Z mod 5" or "Z sub 5") represents the set of integers modulo 5, which forms a group under the operation of addition. In simpler terms, it's a collection of remainders when integers are divided by 5.
The elements of the group $\mathbb{Z}_5$ are:
- 1
- 2
- 3
- 4
When performing addition in $\mathbb{Z}_5$, the result is always reduced modulo 5. For example, $3 + 4 = 7$, which is $2$ modulo 5, so $3 + 4 = 2$ in $\mathbb{Z}_5$.
What is Group Order?
In abstract algebra, the order of a group is simply the number of elements it contains. It's a fundamental characteristic that helps classify and understand different groups. For finite groups, counting the elements directly gives its order.
Determining the Order of $\mathbb{Z}_5$
To find the order of the group $\mathbb{Z}_5$, we count the distinct elements within its set. As listed above, the elements are ${0, 1, 2, 3, 4}$. There are precisely five elements. Therefore, the order of the group $\mathbb{Z}_5$ is 5.
This group is also a cyclic group, as all its elements can be generated by a single element (for example, by 1, since $1+1=2$, $1+1+1=3$, and so on, up to $1+1+1+1+1=0$ modulo 5). Cyclic groups of prime order, like $\mathbb{Z}_5$, have particularly simple and well-understood structures.
Key Properties of $\mathbb{Z}_5$
Beyond its order, $\mathbb{Z}_5$ possesses several other important properties that define its algebraic structure. These properties are summarized below:
| Property | Value |
|---|---|
| Order | 5 |
| Exponent | 5 |
| Derived Length | 1 |
| Frattini Length | 1 |
- Exponent: The exponent of a group is the smallest positive integer $n$ such that $x^n = e$ (the identity element) for all elements $x$ in the group. For $\mathbb{Z}_5$ under addition, this means $n \cdot x = 0 \pmod 5$ for all $x \in {0,1,2,3,4}$. The smallest such $n$ is 5.
- Derived Length: This property relates to the commutativity of a group. A derived length of 1 indicates that the group is abelian (commutative), meaning the order of operations does not affect the result ($a+b = b+a$). $\mathbb{Z}_5$ is indeed an abelian group.
- Frattini Length: This refers to the length of the Frattini series, a sequence of subgroups related to the structure of the group. A Frattini length of 1 for $\mathbb{Z}_5$ reflects its simple and well-structured nature as a cyclic group of prime order.
