Group Homomorphisms

Group homomorphisms are the basic vocabulary of comparison throughout mathematics and its applications. In linear algebra and representation theory, a group representation is by definition a homomorphism $G \to GL_n(\mathbb{C})$, and the kernel measures which group elements act trivially — central to quantum mechanics, where representations of symmetry groups (rotations, Lorentz transformations) classify particle states, and the kernel identifies symmetries that act as pure phase or identity. In number theory and cryptography, homomorphisms between multiplicative groups modulo primes underlie the discrete logarithm problem and Diffie–Hellman key exchange; the difficulty of computing preimages (rather than kernels, which are often trivial there) is what secures the protocol. In coding theory, linear codes are kernels of homomorphisms (parity-check maps) from $\mathbb{F}_2^n$ to $\mathbb{F}_2^k$, and normality of the kernel (automatic since these groups are abelian) underlies the coset-decoding algorithms used to correct errors. In topology, the fundamental group functor sends continuous maps between spaces to group homomorphisms between fundamental groups, and kernels there detect which loops become contractible after mapping — a cornerstone of algebraic topology and covering space theory. In computer science, homomorphic encryption schemes are built from algebraic structures where an operation on ciphertexts corresponds, via a homomorphism, to an operation on plaintexts, allowing computation on encrypted data without decryption.

Ask the tutor: Can you show me, step by step, why the image of a homomorphism need not be normal in the target group, with an explicit nonabelian example?

Think of $\varphi$ as a translator between two languages of group elements: it does not necessarily know every word in $H$ (it need not be surjective), and it may translate several different words of $G$ into the same word of $H$ (it need not be injective). What it must do faithfully is preserve grammar — the order in which you combine elements and then translate matches translating first and then combining. The kernel captures exactly the "silent" elements of $G$ — the words that translate to silence, the identity of $H$. Because $\varphi$ preserves the operation, this silent set cannot be an arbitrary subset: it must be closed under multiplication and inversion (so it is a subgroup), and moreover it must be invariant under conjugation by any element of $G$ (so it is normal). Intuitively, if $k$ is silent, then conjugating $k$ by any $g$ produces something that translates to $\varphi(g)\,e_H\,\varphi(g)^{-1} = e_H$ — still silent. This is the mechanism, not a coincidence, that forces normality. The image, meanwhile, is the "vocabulary actually used" — a subgroup of $H$ that records how much of $H$ is reachable from $G$. The next figure visualizes a concrete homomorphism $\varphi: \mathbb{Z} \to \mathbb{Z}/6\mathbb{Z}$ given by reduction mod 6, restricted for legibility to $\{0,\dots,11\}$ mapping onto the 6 residues, showing fibers collapsing onto single output points.

This corollary is precisely why the theory of kernels and the theory of normal subgroups are two descriptions of the same phenomenon — a fact whose full consequences (reconstructing $\operatorname{im}\varphi$ as $G/\ker\varphi$) are developed under The Isomorphism Theorems.

Example 1 — determinant map on a matrix group. Let $G = GL_2(\mathbb{R})$, the group of invertible $2\times 2$ real matrices under multiplication, and let $H = (\mathbb{R}^\times, \cdot)$, the nonzero reals under multiplication. Define

This is a homomorphism because $\det(AB) = \det(A)\det(B)$ for all $A, B \in GL_2(\mathbb{R})$, a standard fact from linear algebra. Let us verify the theorem's conclusions concretely with an asymmetric example. Take

Then $\det A = 2(3) - 1(5) = 1$ and $\det B = 1(1) - 0(4) = 1$. Compute

so $\det(AB) = 6(3) - 1(17) = 18 - 17 = 1$. Indeed $\det(A)\det(B) = 1 \cdot 1 = 1 = \det(AB)$, consistent with $\varphi(AB)=\varphi(A)\varphi(B)$. The kernel is

the special linear group. By Theorem part (5), $SL_2(\mathbb{R})$ is automatically a normal subgroup of $GL_2(\mathbb{R})$ — a fact that would require a separate (messier) direct conjugation computation without the homomorphism machinery: for $g \in GL_2(\mathbb{R})$ and $A \in SL_2(\mathbb{R})$,

so $gAg^{-1} \in SL_2(\mathbb{R})$, confirming normality directly and matching the general theorem. The image is all of $\mathbb{R}^\times$, since for any nonzero real $c$, the matrix $\operatorname{diag}(c,1)$ has determinant $c$. So $\varphi$ is surjective but (since $SL_2(\mathbb{R})$ is nontrivial) far from injective — its kernel is huge, not $\{I\}$, consistent with Theorem part (6): a nontrivial kernel signals non-injectivity, which is obvious here since $\operatorname{diag}(1,1)$ and $\operatorname{diag}(-1,-1)$ both have determinant $1$ but are different — wait, both are in the kernel, illustrating many-to-one behavior sending distinct kernel elements to the same value $1 = e_H$.

Example 2 — a homomorphism between finite cyclic groups with unequal orders. Let $G = \mathbb{Z}/12\mathbb{Z}$ and $H = \mathbb{Z}/8\mathbb{Z}$, both under addition, written additively. Define

First we must check this is well-defined: if $\bar n = \bar m$ in $\mathbb{Z}/12\mathbb{Z}$, i.e. $n - m = 12t$ for some integer $t$, then

so $4n \equiv 4m \pmod 8$, confirming $\varphi(\bar n)$ does not depend on the representative $n$. It is a homomorphism because

Let's tabulate $\varphi$ on all 12 elements of $G$ (reducing $4n \bmod 8$):

(Since $4n \bmod 8$ cycles as $4 \cdot 0=0$, $4\cdot1=4$, $4\cdot2=8\equiv0$, $4\cdot3=12\equiv4$, and repeats with period 2.) So $\operatorname{im}\varphi = \{\bar 0, \bar 4\} \subset \mathbb{Z}/8\mathbb{Z}$, a subgroup of order 2 — confirming Theorem part (3) (the image is a genuine subgroup, here the unique order-2 subgroup of $\mathbb{Z}/8\mathbb{Z}$). The kernel is

reading off the table for which entries give $0$. This is exactly $2\mathbb{Z}/12\mathbb{Z} = \{0,2,4,6,8,10\}$, a subgroup of order 6 in $G$ (index 2). Since $\mathbb{Z}/12\mathbb{Z}$ is abelian, every subgroup is automatically normal, so Theorem part (5) is not vacuous here but is not the interesting content — the interesting content is that it holds even when normality is not free (as in Example 1, a nonabelian group). Since $\ker\varphi \ne \{\bar 0\}$, Theorem part (6) tells us $\varphi$ is not injective — confirmed directly by the table, e.g. $\varphi(\bar 0) = \varphi(\bar 2) = 0$ despite $\bar 0 \ne \bar 2$.