Show that f(x,y)=x(-1)^y is homomorphism from the direct product group ℝ(positive) ×ℤ2 to the group ℝ*.
(need to show that f((a,b)(c,d))=f(a,b)f(c,d), is it right? but I can't get them equal to each other)
Take $\displaystyle (a,b),(c,d)\in \mathbb{R}^+ \times \mathbb{Z}_2$ then $\displaystyle f((a,b)(c,d))=ac(-1)^{b+d}=ac(-1)^b(-1)^d=a(-1)^bc(-1)^d=f((a,b))f((c,d))$
I'm assuming $\displaystyle \mathbb{R} ^+$ is the group of postitive real numbers with multiplication as operation.