Associativity of $m * n = m+(-1)^m n$ in $\mathbb{Z}$

Question

I have to show that $\mathbb{Z}$ with the operation $m*n = m+(-1)^m n$ is a group (Exercise 1.2.4 from Beardon's Algebra nad Geometry). There must be something wrong with this "proof" of associativity, but I can't find where:

$m*(n*o) = m + (-1)^m(n+(-1)^no) = m + (-1)^mn + (-1)^{m+n}o$; whereas

$(m*n)*o = (m+(-1)^mn)+(-1)^{m+(-1)^mn}o$.

It should be that $m + (-1)^mn + (-1)^{m+n}o = m+(-1)^mn+(-1)^{m+(-1)^mn}o$, but $m+(-1)^mn = m+n$ only when $m$ is even. So, where's the blunder?

Answer 1

Hint: $(-1)^x=(-1)^y\iff x\equiv y\pmod 2$ for integers $x,y$, and not only if $x=y$.