What does $\Bbb Z_7$ mean?

Question

What does $\Bbb Z_7$ mean? Does it mean that it contains $\{0,1,2,3,4,5,6,7\}$? I just want to know what it mean

Answer 1

By some accounts $\mathbb Z_7$ contains just $0,1,2,3,4,5,6,$ but one thinks of those as representing "congruence classes": $$ \begin{array}{c|l} \text{representative} & \text{congruence class} \\ \hline \vphantom{\dfrac {\displaystyle\sum} 1} 0 & \{\,0, \pm7,\pm14,\pm21,\pm28,\pm35,\ldots \,\} \text{ (multiples of 7)} \\[8pt] 1 & \{\, 1, 1\pm7, 1\pm14, 1\pm21, \ldots \,\} = \{\, \ldots, -13,-6,1,8,15,\ldots \,\} \\[8pt] 2 & \{\, 2, 2\pm7, 2\pm14, 2\pm21,\ldots \,\} = \{\,\ldots, -12, -5, 2, 9, 16, \ldots \,\} \\[8pt] 3 & \{\, 3, 3\pm7, 3\pm14, 3\pm21,\ldots \,\} = \{\,\ldots, -11, -4, 3, 10, 17, \ldots \,\} \\[8pt] 4 & \{\, 4, 4\pm7, 4\pm14, 4\pm21,\ldots \,\} = \{\,\ldots, -10, -3, 4, 11, 18, \ldots \,\} \\[8pt] 5 & \{\, 5, 5\pm7, 5\pm14, 5\pm21,\ldots \,\} = \{\,\ldots, -9, -2, 5, 12, 19, \ldots \,\} \\[8pt] 6 & \{\, 6, 6\pm7, 6\pm14, 6\pm21,\ldots \,\} = \{\,\ldots,-15, -8, -1, 6, 13, 16, \ldots \,\} \end{array} $$ If we were to take this table one step further and include $7,$ we would be back to the same congruence class we started with, represented by $0.$

One says that two numbers belonging to the same congruence class are congruent to each other modulo $7.$ For example, $11$ and $18$ are congruent to each other. One writes: $$ (18\equiv 11) \pmod 7. $$ Or one writes $$ 18\equiv 11 \pmod 7. $$ I used parentheses the first time in order to be explicit about the syntax, since sometimes people mistakenly think it means this: $\require{cancel} \xcancel{18 \equiv (11\bmod 7)},$ and that leads to confusions that have been the subject of questions posted here.

One can add, subtract, or multiply congruence classes.

Answer 2

$\Bbb Z_7$ is the set of residue classes modulo $7$. So $$\Bbb Z_7=\{[0]_7,[1]_7,[2]_7,[3]_7,[4]_7,[5]_7,[6]_7\}$$ where $[a]_7$ is the residue class of $a\in\Bbb Z$ modulo $7$. However, we often just write $\Bbb Z_7=\{0,1,2,3,4,5,6\}$ and it is implicit that these are residue classes we are talking about.

Note that $7=0$ in $\Bbb Z_7$, so when you include $7$ and $0$ in your set, they are the same element.

Answer 3

It is the set of natural numbers that leave a remainder when you divide by $7$, or the set of integers modulo $7$ if you want to be a bit more formal. So $$\Bbb Z_7=\{0,1,2,3,4,5,6\}.$$

Note that $7$ is not in this set as having a remainder of $7$ is the same as having a remainder of $0$ when you divide by $7$.

Answer 4

$\mathbb{Z}_7$ usually denotes the group of integers modulo $7$. That means you partition the integers into equivalence classes. You do so by looking at each integer and check what it gives as a remainder when you divide by $7$. From each class we can pick a representative number and we usually choose the smallest from each class. So the members are $\{0,1,2,3,4,5,6\}$. $7$ gives remainder $0$ when you divide by $7$ so it is in the class of $0$.

Answer 5

From the Axioms

Since this question is tagged group-theory, the most appropriate answer is likely that $\mathbb{Z}_7$ is the abelian group with seven elements. Taking this apart a little, $\mathbb{Z}_7$ is a set which contains seven elements. So we could write it as a set as (for example) $$ \{0,1,2,3,4,5,6\} \qquad\text{or}\qquad \{a,b,c,d,e,f,g\} \qquad\text{or}\qquad \{ \ast,\square,\spadesuit,\heartsuit,\diamondsuit,\clubsuit,\nabla \}. $$ In addition to being a set with seven elements, $\mathbb{Z}_7$ also has the structure of an abelian group. This means that there is a binary operation $+$ on $\mathbb{Z}_7$ which satisfies the following axioms:

1. (Closure) If $x$ and $y$ are any two elements of $\mathbb{Z}_7$, then $x+y$ is also an element of $\mathbb{Z}_7$.
2. (Associativity) If $x$, $y$, and $z$ are all elements of $\mathbb{Z}_7$, then $$ (x+y) + z = z + (y+z); $$ that is, we can move parentheses around without penalty.
3. (Identity) There exists a special element $\mathbb{0}$ such that $$ \mathbb{0} + x = x + \mathbb{0} = x $$ for all $x$ in $\mathbb{Z}_7$. This element is called the identity element.
4. (Inverse) If $x$ is an element of $\mathbb{Z}_7$, then there exists an element $-x$ in $\mathbb{Z}_7$ such that $$ x + (-x) = \mathbb{0}. $$ That is, each element of $\mathbb{Z}_7$ has an inverse.
5. (Commutativity) For all $x$ and $y$ in $\mathbb{Z}_7$, we have $$ x + y = y + x. $$ That is, the binary operation is commutative, and we can "add things up" in any order.

Since $\mathbb{Z}_7$ must have a lot of the same properties that we might expect from the integers, it is reasonable to represent it with a set that looks like integers. Thus we can very reasonably write $$ \mathbb{Z}_7 = \{0,1,2,3,4,5,6\} $$ as a set. For the group structure, consider the following addition table: $$ \begin{array}{r|rrrrrrr} + & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\hline 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 0 \\ 2 & 2 & 3 & 4 & 5 & 6 & 0 & 1 \\ 3 & 3 & 4 & 5 & 6 & 0 & 1 & 2 \\ 4 & 4 & 5 & 6 & 0 & 1 & 2 & 3 \\ 5 & 5 & 6 & 0 & 1 & 2 & 3 & 4 \\ 6 & 6 & 0 & 1 & 2 & 3 & 4 & 5 \end{array} $$ Notice that $0 + x = x + 0 = x$ for all $x\in \mathbb{Z}_7$, and that each element has an inverse (this can be seen by noting that there is a zero in each row). Closure should be more-or-less obvious, and associativity and commutativity can be checked by brute force. Thus we can define $\mathbb{Z}_7$ to be the set $\{0,1,2,3,4,5,6\}$ together with the operation $+$ satisfying the above table.

---

As a Quotient Group

A bit more abstractly, we can define $\mathbb{Z}_7$ to be the quotient group of $\mathbb{Z}$ and the subgroup generated by $7$. The actual construction is a little more abstract, but the basic outline is as follows: $\mathbb{Z}$ is an abelian group. We say that two integers are equivalent modulo 7 if their difference is divisible by 7. For example, 23 and 37 are equivalent modulo 7, as $$ 37 - 23 = 14 = 2\cdot 7. $$ For simplicity, we write $$ 37 \equiv 23 \pmod{7}, $$ which is read "37 is equivalent to 23 modulo 7." From this definition, it can be shown that if $x$ is an integer, then the set of integers that are equivalent to $x$ is the set $$ \{ x+7k \mid k\in\mathbb{Z} \}. $$ That is, $x$ plus any integer multiple of $7$ will be equivalent to $x$ modulo 7. This equivalence relation partitions the integers into seven disjoint classes: \begin{align} [0]_7 &:= \{7k \mid k\in\mathbb{Z}\} &&(\text{multiples of 7})\\ [1]_7 &:= \{1+7k \mid k\in\mathbb{Z}\} &&(\text{multiples of 7, plus 1})\\ [2]_7 &:= \{2+7k \mid k\in\mathbb{Z}\} &&(\text{and so on})\\ [3]_7 &:= \{3+7k \mid k\in\mathbb{Z}\} \\ [4]_7 &:= \{4+7k \mid k\in\mathbb{Z}\} \\ [5]_7 &:= \{5+7k \mid k\in\mathbb{Z}\} \\ [6]_7 &:= \{6+7k \mid k\in\mathbb{Z}\} \end{align} Next, observe that if $x \in [a]_7$ and $y\in [b]_7$, then $$ (x+y) \in [a+b]_7. $$ It then follows that the set $\{ [0]_7, [1]_7, \dotsc, [6]_7\}$, together with the binary operation $\oplus$ defined by $$ [a]_7 \oplus [b]_7 := [a+b]_7, $$ satisfy the axioms of an abelian group. We are using the funny addition symbol $\oplus$ here in order to distinguish the addition of equivalence classes from the usual addition of integers. The addition within square braces (i.e. in the term $[a+b]_7$) is the usual addition. In this construction, we are starting with a "big" group (the integers), then "modding out" by a subgroup (the multiples of 7). In notation, we might write this as $$ \mathbb{Z}/\langle 7 \rangle \qquad\text{or}\qquad \mathbb{Z}/7\mathbb{Z} \qquad\text{or}\qquad \mathbb{Z}/7. $$ The above is a little hand-wavy, but it can be made rigorous. Note that $\mathbb{Z}/\langle 7 \rangle$ is an abelian group with 7 elements. It is natural to ask if it is the "same" as the group defined previously.

Indeed, if $\{0,1,\dotsc,6\}$ is the set of seven elements with addition as defined in the table above, the map $$ \varphi: \{0,1,2,3,4,5,6\} \to \{ [0]_7,[1]_7,[2]_7,[3]_7,[4]_7,[5]_7,[6]_7\} $$ defined by $\varphi(a) = [a]_7$ is an isomorphism of groups. This means that the groups $$ (\{0,1,2,3,4,5,6\}, +) \qquad\text{and}\qquad ( \{ [0]_7,[1]_7,[2]_7,[3]_7,[4]_7,[5]_7,[6]_7\}, \oplus ) $$ cannot be distinguished from each other. Huzzah!

---

And Now for Something Completely Different

In addition to the above, the notation $\mathbb{Z}_7$ is also used to denote the $7$-adic integers. Because I am analyst (and not an algebraist), I'll describe the metric spaces version of the $7$-adic numbers, though do note that they can also be constructed in a more algebra-flavoured manner (as the inverse limit of some sequence of rings, plus a little more abstract nonsense).

If $x$ is any rational number, then there is a unique integer (which could be negative, or zero) such that $$ x = 7^n \frac{a}{b}, $$ where $a$ and $b$ are integers which are not divisible by $7$. We define the $7$-adic absolute value of $x$ to be $$ |x|_7 := 7^{-n}. $$ Basically, the $7$-adic absolute value measure how "divisible" a rational number is by $7$. The more divisible it is, the smaller it is. More generally, if $p$ is any prime number, we can define a $p$-adic absolute value which measures how divisible rational numbers are by $p$. In some sense, the $p$-adic absolute value measures the "$p$-ness" of a rational number (let that one sink in for a moment).

The $7$-adic absolute value gives us a way of describing how big or small rational numbers are, which can be used to determine a distance between two rational numbers. Specifically, the function $$ d_7(x,y) := |x-y|_7 $$ defines a metric on $\mathbb{Q}$, analogous to the way that the "usual" absolute value can be used to define a metric. As with the usual absolute value, the metric obtained from the $7$-adic absolute value does not turn the rational numbers into a complete metric space: there are sequences of rational numbers that get "close together" (in a certain technical sense), but which do not converge to a rational limit. However, we can "complete" the rational numbers using a relatively straightforward technique. Once we do this, we obtain a set denoted $\mathbb{Q}_7$, called the $7$-adic numbers. (Again, by analogy, the usual absolute value on $\mathbb{Q}$ induced an incomplete metric which can be completed in order to obtain the real numbers.)

In the $7$-adic numbers, the unit ball (the set of $7$-adic numbers less than or equal one unit from zero) plays a special role. From a metric point of view, a distinguished ball is nice to have, but it turns out that this unit ball also has really nice algebraic structure. Because of this, it gets special notation: $$ \mathbb{Z}_7 = B_{\le}(0,1) = \{ x\in\mathbb{Q}_p \mid |x|_7 \le 1 \}. $$ Note that this unit ball—called the $7$-adic integers (for reasons that are more clear if you construct the $7$-adic numbers algebraically, rather than analytically)—is also denoted by the symbol $\mathbb{Z}_7$. However, this object is very different from the group $\mathbb{Z}/\langle 7 \rangle$, thus context is important.