7.1: Homomorphisms
Let \(G_1\) and \(G_2\) be groups. Recall that \(\phi:G_1\to G_2\) is an isomorphism if and only if \(\phi\)
- is one-to-one,
- is onto, and
- satisfies the homomorphic property.
We say that \(G_1\) is isomorphic to \(G_2\) and write \(G_1\cong G_2\) if such a \(\phi\) exists. Loosely speaking, two groups are isomorphic if they have the “same structure." What if we drop the one-to-one and onto requirement?
Let \((G_1,*)\) and \((G_2,\odot)\) be groups. A function \(\phi:G_1\to G_2\) is a homomorphism if and only if \(\phi\) satisfies the homomorphic property: \[\phi(x*y)=\phi(x)\odot\phi(y)\] for all \(x,y\in G_1\) . At the risk of introducing ambiguity, we will usually omit making explicit reference to the binary operations and write the homomorphic property as \[\phi(xy)=\phi(x)\phi(y).\]
Group homomorphisms are analogous to linear transformations on vector spaces that one encounters in linear algebra.
Figure \(\PageIndex{1}\) captures a visual representation of the homomorphic property. We encountered this same representation in Figure \(\PageIndex{1}\). If \(\phi(x)=x'\) , \(\phi(y)=y'\) , and \(\phi(z)=z'\) while \(z'=x'\odot y'\) , then the only way \(G_2\) may respect the structure of \(G_1\) is for \[\phi(x*y)=\phi(z)=z'=x'\odot y'=\phi(x)\odot \phi(y).\]
Define \(\phi:\mathbb{Z}_3\to D_3\) via \(\phi(k)=r^k\) . Prove that \(\phi\) is a homomorphism and then determine whether \(\phi\) is one-to-one or onto. Also, try to draw a picture of the homomorphism in terms of Cayley diagrams.
Let \(G\) and \(H\) be groups. Prove that the function \(\phi:G\times H\to G\) given by \(\phi(g,h)=g\) is a homomorphism. This function is an example of a projection map .
There is always at least one homomorphism between two groups.
Let \(G_1\) and \(G_2\) be groups. Define \(\phi:G_1\to G_2\) via \(\phi(g)=e_2\) (where \(e_2\) is the identity of \(G_2\) ). Then \(\phi\) is a homomorphism. This function is often referred to as the trivial homomorphism or the \(0\) -map .
Back in Section 3.3 , we encountered several theorems about isomorphisms. However, at the end of that section we remarked that some of those theorems did not require that the function be one-to-one and onto. We collect those results here for convenience.
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism.
- If \(e_1\) and \(e_2\) are the identity elements of \(G_1\) and \(G_2\) , respectively, then \(\phi(e_1)=e_2\) .
- For all \(g\in G_1\) , we have \(\phi(g^{-1})=[\phi(g)]^{-1}\) .
- If \(H\leq G_1\) , then \(\phi(H)\leq G_2\) , where \[\phi(H):=\{y\in G_2\mid \text{there exists } h\in H\text{ such that }\phi(h)=y\}.\] Note that \(\phi(H)\) is called the image of \(H\) . A special case is when \(H=G_1\) . Notice that \(\phi\) is onto exactly when \(\phi(G_1)=G_2\) .
The following theorem is a consequence of Lagrange’s Theorem.
Let \(G_1\) and \(G_2\) be groups such that \(G_2\) is finite and let \(H\leq G_1\) . If \(\phi:G_1\to G_2\) is a homomorphism, then \(|\phi(H)|\) divides \(|G_2|\) .
The next theorem tells us that under a homomorphism, the order of the image of an element must divide the order of the pre-image of that element.
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. If \(g\in G_1\) such that \(|g|\) is finite, then \(|\phi(g)|\) divides \(|g|\) .
Every homomorphism has an important subset of the domain associated with it.
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. The kernel of \(\phi\) is defined via \[\ker(\phi):=\{g\in G_1\mid \phi(g)=e_2\}.\]
The kernel of a homomorphism is analogous to the null space of a linear transformation of vector spaces.
Identify the kernel and image for the homomorphism given in Problem \(\PageIndex{1}\).
What is the kernel of a trivial homomorphism (see Theorem \(\PageIndex{1}\)).
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. Then \(\ker(\phi)\trianglelefteq G_1\) .
Let \(G\) be a group and let \(H\trianglelefteq G\) . Then the map \(\gamma:G\to G/H\) given by \(\gamma(g)=gH\) is a homomorphism with \(\ker(\gamma)=H\) . This map is called the canonical projection map .
The upshot of Theorems \(\PageIndex{5}\) and \(\PageIndex{6}\) is that kernels of homomorphisms are always normal and every normal subgroup is the kernel of some homomorphism. It turns out that the kernel can tell us whether \(\phi\) is one-to-one.
The next theorem tells us that two elements in the domain of a group homomorphism map to the same element in the codomain if and only if they are in the same coset of the kernel.
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. Then \(\phi(a) = \phi(b)\) if and only if \(a\in b\ker(\phi)\) .
One consequence of Theorem \(\PageIndex{7}\) is that if the kernel of a homomorphism has order \(k\) , then the homomorphism is \(k\) -to-1. That is, every element in the range has exactly \(k\) elements from the domain that map to it. In particular, each of these collections of \(k\) elements corresponds to a coset of the kernel.
Suppose \(\phi:\mathbb{Z}_{20}\to\mathbb{Z}_{20}\) is a group homomorphism such that \(\ker(\phi)=\{0,5,10,15\}\) . If \(\phi(13)=8\) , determine all elements that \(\phi\) maps to 8.
The next result is a special case of Theorem \(\PageIndex{7}\).
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. Then \(\phi\) is one-to-one if and only if \(\ker(\phi)=\{e_1\}\) , where \(e_1\) is the identity in \(G_1\) .
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. Given a generating set for \(G_1\) , the homomorphism \(\phi\) is uniquely determined by its action on the generating set for \(G_1\) . In particular, if you have a word for a group element written in terms of the generators, just apply the homomorphic property to the word to find the image of the corresponding group element.
Suppose \(\phi: Q_8\to V_{4}\) is a group homomorphism satisfying \(\phi(i)=h\) and \(\phi(j)=v\) .
- Find \(\phi(1)\) , \(\phi(-1)\) , \(\phi(k)\) , \(\phi(-i)\) , \(\phi(-j)\) , and \(\phi(-k)\) .
- Find \(\ker(\phi)\) .
- What well-known group is \(Q_8/\ker(\phi)\) isomorphic to?
Find a non-trivial homomorphism from \(\mathbb{Z}_{10}\) to \(\mathbb{Z}_6\) .
Find all non-trivial homomorphisms from \(\mathbb{Z}_3\) to \(\mathbb{Z}_6\) .
Prove that the only homomorphism from \(D_3\) to \(\mathbb{Z}_3\) is the trivial homomorphism.
Let \(F\) be the set of all functions from \(\mathbb{R}\) to \(\mathbb{R}\) and let \(D\) be the subset of differentiable functions on \(\mathbb{R}\) . It turns out that \(F\) is a group under addition of functions and \(D\) is a subgroup of \(F\) (you do not need to prove this). Define \(\phi:D\to F\) via \(\phi(f)=f'\) (where \(f'\) is the derivative of \(f\) ). Prove that \(\phi\) is a homomorphism. You may recall facts from calculus without proving them. Is \(\phi\) one-to-one? Onto?