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)

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- Nov 21st 2009, 08:51 PMapple2009homomorphism
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) - Nov 21st 2009, 09:38 PMJose27
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.