Exercise 974Algebra › GroupsL1Added 3 Nov 2026

Homomorphisms between familiar groups

#homomorphism#examples#kernels

Statement

For each map, determine whether it is a group homomorphism, and if so, compute its kernel and image:

(a) $f:(\mathbb{Z},+)\to(\mathbb{Z}/n\mathbb{Z},+)$ , $k\mapsto\bar k$ ;
(b) $g:(\mathbb{R},+)\to(\mathbb{R}^*,\times)$ , $x\mapsto e^x$ ;
(c) $h:(\mathbb{C}^*,\times)\to(\mathbb{R}^*_+,\times)$ , $z\mapsto|z|$ ;
(d) $\varphi:(\mathbb{R},+)\to(\mathbb{R},+)$ , $x\mapsto x^2$ .

Hint

Check $f(ab)=f(a)f(b)$ (or the additive analogue).

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Solution

(a) $f(k+\ell)=\overline{k+\ell}=\bar k+\bar\ell=f(k)+f(\ell)$ . Homomorphism. Kernel $n\mathbb{Z}$ , image $\mathbb{Z}/n\mathbb{Z}$ (surjective).

(b) $g(x+y)=e^{x+y}=e^xe^y=g(x)g(y)$ . Homomorphism. Kernel $\{0\}$ (injective), image $\mathbb{R}^*_+$ .

(c) $h(zw)=|zw|=|z||w|=h(z)h(w)$ . Homomorphism. Kernel $\{z:|z|=1\}$ (unit circle), image $\mathbb{R}^*_+$ (surjective).

(d) $\varphi(x+y)=(x+y)^2\neq x^2+y^2$ in general. Not a homomorphism (unless the group operation is trivial).

Conclusion. (a), (b), (c) are homomorphisms; (d) is not.

Homomorphisms between familiar groups

Statement

Solution

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