The order of the group $\mathbb{Z}_7$ is 7.
Understanding the Group $\mathbb{Z}_7$
The group $\mathbb{Z}_7$, often denoted as Z7, represents the cyclic group of order 7. This group consists of the integers modulo 7 under the operation of addition. Its elements are ${0, 1, 2, 3, 4, 5, 6}$.
The order of a group refers to the number of elements it contains. In the case of $\mathbb{Z}_7$, there are precisely 7 distinct elements, which directly corresponds to its order.
Key Properties of $\mathbb{Z}_7$
As a finite cyclic group, $\mathbb{Z}_7$ exhibits several fundamental properties that are characteristic of such structures. Here's a summary of its key attributes:
| Property | Value |
|---|---|
| Order | 7 |
| Exponent | 7 |
| Frattini Length | 1 |
| Fitting Length | 1 |
- Order: As established, the group $\mathbb{Z}_7$ has 7 elements.
- Exponent: The exponent of a group is the smallest positive integer $n$ such that $g^n = e$ (the identity element) for every element $g$ in the group. For $\mathbb{Z}_7$, since it is a cyclic group of prime order, its exponent is equal to its order, which is 7.
- Frattini Length & Fitting Length: Both the Frattini length and Fitting length are 1 for $\mathbb{Z}_7$. These properties relate to the subgroup structure and solvability of the group, indicating its relatively simple and well-behaved nature as a finite cyclic group.
Every element in $\mathbb{Z}_7$ (except for the identity element 0) generates the entire group, meaning it can produce all other elements through repeated addition modulo 7. For example, starting with 1:
- $1 \pmod 7$
- $1+1 = 2 \pmod 7$
- $1+1+1 = 3 \pmod 7$
- ...
- $1+1+1+1+1+1+1 = 7 \equiv 0 \pmod 7$ (the identity)
This confirms that the group has 7 distinct elements before reaching the identity again.
