Group homomorphism

Respecting the operation means it does not matter whether we operate before or after applying \(\varphi\). Examples: \(n\mapsto n\bmod m\) from \((\mathbb{Z},+)\) to \((\mathbb{Z}_m,+)\); the exponential from \((\mathbb{R},+)\) to \((\mathbb{R}^*,\cdot)\); and the determinant from \(GL_n(\mathbb{R})\) to \(\mathbb{R}^*\). The constant function to the identity is also one, called the trivial homomorphism.

The seemingly weak condition forces more: \(\varphi(e_G)=e_H\) and \(\varphi(g^{-1})=\varphi(g)^{-1}\). It suffices to note that \(\varphi(e_G)\varphi(e_G)=\varphi(e_G)\) and cancel. That is why identities and inverses need not be required explicitly.

The image \(\varphi(G)\) is a subgroup of \(H\), and the composition of homomorphisms is a homomorphism. A homomorphism from \(\mathbb{Z}_m\) to any group is determined by the image \(a\) of \(1\), which must satisfy \(a^m=e\); hence there are \(\gcd(m,n)\) homomorphisms from \(\mathbb{Z}_m\) to \(\mathbb{Z}_n\).

Typical mistake: believing that any function that \"looks like\" addition works. \(n\mapsto n+1\) is not one, since it sends \(0\) to \(1\neq e\). When the function is also bijective we reach the isomorphism, and the set of elements that \(\varphi\) sends to the identity, the kernel, is the next object to study.