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text: '[summary: A group homomorphism is a function between [3gd groups] which "respects the group structure".]\n\n[summary(Technical): Formally, given two groups $(G, +)$ and $(H, *)$ (which hereafter we will abbreviate as $G$ and $H$ respectively), a group homomorphism from $G$ to $H$ is a [-3jy] $f$ from the underlying set $G$ to the underlying set $H$, such that $f(a) * f(b) = f(a+b)$ for all $a, b \\in G$.]\n\nA group homomorphism is a function between [3gd groups] which "respects the group structure".\n\n#Definition\n\nFormally, given two groups $(G, +)$ and $(H, *)$ (which hereafter we will abbreviate as $G$ and $H$ respectively), a group homomorphism from $G$ to $H$ is a [-3jy] $f$ from the underlying set $G$ to the underlying set $H$, such that $f(a) * f(b) = f(a+b)$ for all $a, b \\in G$.\n\n#Examples\n\n - For any group $G$, there is a group homomorphism $1_G: G \\to G$, given by $1_G(g) = g$ for all $g \\in G$. This homomorphism is always [499 bijective].\n - For any group $G$, there is a (unique) group homomorphism into the group $\\{ e \\}$ with one element and the only possible group operation $e * e = e$. This homomorphism is given by $g \\mapsto e$ for all $g \\in G$. This homomorphism is usually not [4b7 injective]: it is injective if and only if $G$ is the group with one element. (Uniqueness is guaranteed because there is only one *function*, let alone group homomorphism, from any set $X$ to a set with one element.)\n - For any group $G$, there is a (unique) group homomorphism from the group with one element into $G$, given by $e \\mapsto e_G$, the identity of $G$. This homomorphism is usually not [4bg surjective]: it is surjective if and only if $G$ is the group with one element. (Uniqueness is guaranteed this time by the property proved below that the identity gets mapped to the identity.)\n - For any group $(G, +)$, there is a bijective group homomorphism to another group $G^{\\mathrm{op}}$ given by taking inverses: $g \\mapsto g^{-1}$. The group $G^{\\mathrm{op}}$ is defined to have underlying set equal to that of $G$, and group operation $g +_{\\mathrm{op}} h := h + g$.\n - For any pair of groups $G, H$, there is a homomorphism between $G$ and $H$ given by $g \\mapsto e_H$.\n - There is only one homomorphism between the group $C_2 = \\{ e_{C_2}, g \\}$ with two elements and the group $C_3 = \\{e_{C_3}, h, h^2 \\}$ with three elements; it is given by $e_{C_2} \\mapsto e_{C_3}, g \\mapsto e_{C_3}$. For example, the function $f: C_2 \\to C_3$ given by $e_{C_2} \\mapsto e_{C_3}, g \\mapsto h$ is *not* a group homomorphism, because if it were, then $e_{C_3} = f(e_{C_2}) = f(gg) = f(g) f(g) = h h = h^2$, which is not true. (We have used that the identity gets mapped to the identity.)\n\n# Properties\n\n- The identity gets mapped to the identity. ([49z Proof.])\n- The inverse of the image is the image of the inverse. ([4b1 Proof.])\n- The [3lh image] of a group under a homomorphism is another group. ([4b4 Proof.])\n- The composition of two homomorphisms is a homomorphism. ([4b6 Proof.])',
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