I was working my way through some basic number theory problems, when in the chapter on "Introduction to Group Theory," I came across the following:

  • Show that for every positive integer $n$, the complete residue class modulo $n$ forms a group under the binary operation of the usual arithmetical addition modulo $n$.
  • Identify the identity element and the inverse of the element $x \pmod n$.

I am a newbie to group theory and am all thumbs, can someone help me out? Perhaps a hint…

  • $\begingroup$ And after you have answered that, consider the non-zero elements with the operation of multiplication (instead of addition). For which n do these elements form a group? $\endgroup$ – marty cohen Apr 13 '15 at 2:15

A group is a set with an associated operation satisfying a certain short list of properties. With this sort of introductory problem, it can be a good idea to explicitly verify the axioms of a group in order to bolster your understanding of what groups are.

Fix $n$ and look at the reduced residue system of integers mod $n$. For concreteness, we'll use the notation $[0]$ to denote the residue class corresponding to residue $0$, and similarly for $[j]$. So for instance, the regular integers $1, n+1, 2n+1$, and so on are all representatives for the residue class $[1]$.

A group needs to have an associative binary operation. The claim here is that integer arithmetic modulo $n$ gives an associative binary operation.

  1. If $[a]$ and $[b]$ are two residue classes, then $[a]+[b] = [a+b \bmod n]$ is another residue class. Indeed, if $a + jn$ is a representative for $[a]$ and $b + kn$ is a representative for $[b]$, then $a + b + n(j+k)$ is a representative for $[a + b \bmod n]$. So addition modulo $n$ is well-defined on residue classes.

  2. Similarly, you can show that addition is associative as normal additional on the integers is associative.

A group needs to have an identity element. Here, you should show that the identity element is the residue class $[0]$.

Each element $[k]$ in our group needs to have an additive inverse $[\overline{k}]$ satisfying $[k] + [\overline{k}] = [0]$. Here, you should show that the inverse of $[k]$ is $[-k] = [n-k]$.

Once you've shown these properties, you will have shown that a complete residue system mod $n$ with addition mod $n$ is a group.

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