Cyclic groups: What is the difference between $\mathbb{Z}_n$, $\mathbb{Z}_n^*$ and $\mathbb{Z}_n^+$?

Question

What is the difference between $\mathbb{Z}_n$, $\mathbb{Z}_n^*$ and $\mathbb{Z}_n^+$? As far as I know, + is a group with generators that generates the entire set $0 \geq x < n $ by addition (and the multiplication is with the * group). But I don't get what $Z_n$ is.

Answer 1

Ugh... Well, for starters, $\mathbb{Z}_n$ is not terribly good notation (I know a lot of books use it; but my abstract algebra course was taught by a number theorist who hates the notation because it clashes with the $p$-adic numbers, and my advisor is a ring theorist who hates it because it clashes with the ring theoretic notation of considering it as ring of integers modulo $n$).

The set $\mathbb{Z}_n$ is the quotient of the group/ring $\mathbb{Z}$ modulo the subgroup/ideal $n\mathbb{Z}$. It is the group (or ring) of “integers modulo $n$.” You can view it either as the set $\{0,1,\ldots,n-1\}$ (or some other set of coset representatives), with multiplication and addition done “modulo $n$” (so $a+b$ is the usual sum $a+b$ if $a+b\lt n$, but it equals $a+b-n$ if $a+b\geq n$; and $ab$ is the remainder when you divide the usual product of $a$ and $b$ by $n$). Or you can view it as a set of congruence classes, $0+n\mathbb{Z},1+n\mathbb{Z},\ldots,(n-1)+n\mathbb{Z}$.

So...

• $\mathbb{Z}_n$ will mean either the ring of integers modulo $n$, or the additive group of integers modulo $n$ (which is a cyclic group of order $n$, in additive notation). Which one should be clear from context.

• $\mathbb{Z}^*_n$ is notation borrowed from ring theory. It means the multiplicative group of units modulo $n$. This is the group of all integers modulo $n$ that are relatively prime to $n$, which form a group under multiplication. For example, $\mathbb{Z}^*_6$ contains only the classes of $1$ and $5$ (the only integers among $0,1,2,3,4,5$ that are relatively prime to $6$), while $\mathbb{Z}^*_8$ contains the classes of $1$, $3$, $5$, and $7$. When $p$ is prime, $\mathbb{Z}^*_p$ contains $1,\ldots,p-1$. This is sometimes called the “(multiplicative) group of units modulo $n$”.

• $\mathbb{Z}_n^+$ means the additive group of integers modulo $n$: it’s like $\mathbb{Z}_n$, but the $+$ superscript ensures there is no possible confusion about whether you are considering the additive group or the ring structure.

Note that $\mathbb{Z}_n^*$ is not always cyclic; it is cyclic if and only if $n$ is $1$, an odd prime power, twice an odd prime power, $2$, or $4$.

Answer 2

In the context of $\textit{groups}$:

$$\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z} =\{[0],[1],...,[n-1] \},$$ where $[x]=\{y\in \mathbb{Z} \ : \ n \ |\ (x-y)\},$ is a $\textit{set}$ of equivalence classes;

$$\mathbb{Z}_n^{+} = (\mathbb{Z}_n,+,[0]),$$ where $[x]+[y] = [x+y],$ is a $\textit{cyclic group}$, generated by $[1]$ for example;

$$\mathbb{Z}_n^{*} = (\{[x] \in \mathbb{Z}_n\ :\ \gcd(x,n)=1 \},*,[1]),$$ where $[x]*[y] = [xy],$ is a $\textit{group}$, not cyclic in general.