(Abstract Algebra) Group Structure Axiom Question

Question

For my homework, I have been asked to analyze the following pair of set and binary operation:

N = The set of integers, subtraction.

So I'm trying to figure out what the identity is for this set/operation...but I can't think of any. There must exist some identity element e in N such that for any x in N, e * x = x * e = x, where * is the generic binary operator, right?

I tried e = 0, but x - 0 is not the same as 0 - x, so this cannot possibly be it... right? Does this mean there does not exist an identity?

Answer 1

Note, firstly, that the set $\mathbb N$ of natural numbers is not closed under subtraction:

$1\in \mathbb N$, $3 \in \mathbb N$, but $\;1- 3\notin\mathbb N$.

So we cannot have a group given subtraction as the binary operation on $\mathbb N$. Nor is subtraction associative: $$a - (b - c) \neq (a-b)-c$$

Answer 2

Note that usually integers are denoted by $\mathbb{Z}$, while $\mathbb{N}$ denotes positive integers. The pair $(\mathbb{Z},-)$ is not a group: for the reason you said and also because subtraction is not associative. Instead $(\mathbb{Z},+)$ is a group.

Answer 3

There are numerous group-theoretic obstructions, e.g. $\,\Bbb N$ is not closed under subtraction, and subtraction is not associative, etc. If you pass to $\,\Bbb Z\,$ then you're on target $\, x = x-e\,\Rightarrow\ e= 0,\,$ hence $x = e-x = -x\,$ a contradiction for $\,x \ne 0.$