What is the meaning of $\mathbb{Z}_{5}^{+}$ (and $\mathbb{Z}_{5}^{*}$) in group theory?

Question

What does $\mathbb{Z}_{5}^{+}$ mean? I know $\mathbb{Z}_{5}$ represents the set of integers modulo 5. I would assume this would mean it is the set of integers modulo 5 under addition except that normally this is notated as $(\mathbb{Z}_{5},+)$. I'm confused as to the meaning of this notation?

Edit:

Furthermore, what is the meaning of $\mathbb{Z}_{5}^{*}$. The context I saw this in seems to imply this is a group but if it was the same thing as $(\mathbb{Z}_{5},*)$ it couldn't be since $(\mathbb{Z}_{5},*)$ lacks some inverses.

Answer 1

For $p$ prime, $\mathbb Z_p^*$ is used to denote the multiplicative group of nonzero elements of $\mathbb Z_p$.

On the other hand, $\mathbb Z_n^×$ denotes the group of units modulo $n$, which has order $\varphi (n) $, where $\varphi$ is Euler's totient function.

For prime $p$, we have $\varphi (p)=p-1$, and indeed $\mathbb Z_p^*=\mathbb Z_p^×$.

Answer 2

The notation $\mathbb{Z}_n^{\times}$ is sometimes used to denote the group of units modulo $n$ with respect to multiplication, so I would presume that $\mathbb{Z}_5^+$ denotes the additive group of integers modulo $5$, in order to contrast it with the multiplicative group.

[For what it's worth, I'll point out that $\mathbb{Z}_5^{\times} \cong \mathbb{Z}_4^+$.]