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In elementary number theory, I find the following notations being used interchangeably, which leads me to have many ambiguous assumptions:

  1. $\mathbb{Z}_p^\times$;
  2. $\mathbb{Z}_p$,

where p is any integer. What's the difference between them?

One more question to add to the kitty: what do they mean when they say "a subgroup of $\mathbb{Z}_p$, where p is prime"?

It will be nice if someone can enlighten me on this elementary notation.

Thanks.

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2 Answers 2

up vote 8 down vote accepted

Usually the notation $\mathbb{Z}_p$ or $\mathbb{Z}/p\mathbb{Z}$ mean the integers modulo $p$, that is $\{0,\ldots,p-1\}$ where you add and multiply as usually and then take the reminder modulo $p$.

The notation of $\mathbb{Z}_p^\times$ is for those numbers which have a multiplicative inverse modulo $p$, namely all $n$ such that exists $m$ such that $n\cdot m$ is $1$ modulo $p$. These are the numbers coprime to $p$, the greatest common divisor of them and $p$ is $1$.

A subgroup of $\mathbb{Z}_p^\times$ is a subset of these numbers which is closed under multiplication (but not necessarily addition), and every number in this subset also has its multiplicative inverse in there.

For example,

$\mathbb{Z}_5 = \{0,1,2,3,4\}$
$\mathbb{Z}_5^\times = \{1,2,3,4\}$
$\{1,4\}$ is a subgroup of $\mathbb{Z}_5^\times$. Can you see why?

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Crystal Clear. :) This is because the multiplicative inverse of 4 is one. A small doubt.. But does that mean a Sub group of Z<sub>p</sub><sup>X</sup> always have an even number of elements since they always contain a number and its inverse? –  bala maverick Mar 7 '11 at 8:23
    
@bala, actually the multiplicative inverse of 4 is 4. –  Asaf Karagila Mar 7 '11 at 8:25
    
Oops.. I overlooked.. Its clear now. Thanks!! :) –  bala maverick Mar 7 '11 at 8:26
2  
@bala: the trivial group $Z^*_2$ = {1} has an odd number of elements. All other multiplicative groups $Z^*_n$ have an even number, but not for the reason you state (an element can be its own inverse!). It will be even because the number of integers coprime to $n$ will be even (see Euler's totient function). –  Fixee Mar 7 '11 at 8:31
    
But of course a $\it subgroup$ of ${\bf Z}_n^{\times}$ can have odd order, even if it's not the trivial subgroup. For example, $\lbrace1,2,4\rbrace$ is a subgroup of ${\bf Z}_7^{\times}$. –  Gerry Myerson May 9 '11 at 3:23

$\mathbb{Z}_p$ is, in my opinion, problematic notion. In slightly less elementary number theory it refers to the p-adic numbers, which are very different from the integers $\bmod p$. An unambiguous, if slightly more cumbersome, notation would be $\mathbb{Z}/(p)$ or $\mathbb{Z}/p\mathbb{Z}$.

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1  
I agree that the use of ${\Bbb Z}_p$ for the integer classes modulo $p$ is somewhat unfortunate since it conflicts with the $p$-adic integers. On the other hand, I always found the notation ${\Bbb Z}/{\Bbb Z}p$ hard to justify, because modular arithmetic is introduced before the general theory of group and ring quotients. –  Andrea Mori Mar 7 '11 at 14:28
2  
I suppose that someone who knows too much ring theory could also (mis)interpret ${\bf Z}_p$ to mean the localization of $\bf Z$ at $p$. –  Gerry Myerson May 9 '11 at 3:25

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