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My notes mention the $U(15)$ group elements as $\{1,2,4,7,8,11,13,14\}$ under multiplication modulo $15$. Someone, please explain how did we reach those elements. The doubt seems to be very basic given the fact that the notes do not elaborate on how to write that specific set.

HeroZhang001
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Pranav Shirke
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    I am unfamiliar with the notation $U(15)$, but it seems you are describing the unit group of the ring $\mathbb{Z}/15\mathbb{Z}$, which is those elements $x$ in $\mathbb{Z}/15\mathbb{Z}={1,2,\dots,15}$ such that $\mathrm{gcd}(x,15)=1$. – student91 Feb 16 '23 at 10:20
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    Welcome to MSE. How do you define the group $U(n)$? – José Carlos Santos Feb 16 '23 at 10:20
  • Yes, usually, $U(n)$ is the unit group of the ring $\Bbb Z/n\Bbb Z$, so $U(n)\cong (\Bbb Z/n\Bbb Z)^{\times}$, the "prime residue group". It has $\phi(n)$ elements. – Dietrich Burde Feb 16 '23 at 10:26
  • @DietrichBurde Where I come from $U(n)$ is unitary group of dimension $n$ and $(\mathbb{Z}/n\mathbb{Z})^*$ is written just like I did. – student91 Feb 16 '23 at 10:28
  • @student91 Yes, where I come from this is also the same as for you (German notation). In the U.S. however, $U_n$ or $U(n)$ can denote both, either unitary group, or prime residue group. I have seen both here at MSE - for example here. The title says $U(n)$, the duplicate then $U_n$. – Dietrich Burde Feb 16 '23 at 10:29
  • Cool, thank you! – student91 Feb 16 '23 at 10:32
  • Pranav, the answer is given here. By definition, the group of units consists of the classes $[k]$, with $\gcd(k,15)=1$, so $k=1,2,4,7,8,11,13,14$ as you said. – Dietrich Burde Feb 16 '23 at 10:34
  • $U(N)$ or $U_N$ are standard notations for the group of units modulo $N$, i.e. the multiplicative group of invertible elements in $\mathbb{Z}/N\mathbb{Z}$. It certainly may arise some confusion with the unitary group, but which is which should be clear from context. (Un)fortunately there are way more meaningful mathematical objects and concept than likely symbols you can form with few letters. – Andrea Mori Feb 16 '23 at 13:42

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Definition: $U(n)$ is the set of all positive integers less than $n$ and relatively prime to $n$ under multiplication modulo $n$ it forms an (Abelian) group, that is $$U(n)=\{x \in \mathbb{N} : 1 \leq x \leq n, gcd(x,n)=1\}$$

if $n=15, U(n)=\{1,2,4,7,8,11,13,14\}$

For some interesting properties of $U(n)$ Contemporary Abstract Algebra by "Joseph A.Gallian" is a good reference book.

Manish Saini
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$\mathbb Z / 15\mathbb Z$ is a ring, so when you examine the (possible) invertible elements, you are talking about the unitgroup or multiplicative group. To do so, you need to ensure that $\operatorname{gcd}(n, 15) = 1$, where $n\in \mathbb Z / 15\mathbb Z$. Since $15 = 3\cdot 5$, we can remove $3,6,9,12,15$ and $5,10,15$, which means we get the set $\{1,2,4,7,8,11,13,14\} $. This is often noted as $(\mathbb Z / 15\mathbb Z)^* = \{n\in\mathbb Z / 15\mathbb Z | \operatorname{gcd}(n, 15) = 1\}.$ I know the notation $U(\mathbb Z / 15\mathbb Z)$ also, but I have not seen $U(15)$. By the way, the cardinality is called the totient.

Edit: Seems that the notation $U_{15}$ is used, too. So best way is always to write what you mean, if the notation is not too common.

mathquester
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