Applications of Reflections: Taking a Group to its “Abelian” Form

Applications of Reflections: Taking a Group to its “Abelian” Form

In continuing the exploration of explicit applications and examples of category-theoretic concepts, we highlight the versatility of reflections and reflective subcategories. This concept can be used to perform all kinds of desired actions on a category to yield a subcategory that “nicer” in some way. This article explores how we can use reflections to make groups abelian.

Please see the previous article by Traylor and Fadeev for the definition of a category and other properties.

Definition (Subcategory):

A category \mathbf{A} is a subcategory of a category \mathbf{B} if 
(1) \mathrm{Ob}(\mathbf{A}) \subseteq \mathrm{Ob}(\mathbf{B})
(2) for each A, A' \in \mathrm{Ob}(\mathbf{A}), \mathrm{hom}_{\mathbf{A}}(A,A') \subseteq \mathrm{hom}_{\mathbf{B}}(A,A')
(3) for each A \in \mathrm{Ob}(\mathbf{A}), the \mathbf{B}-identity on A is the same as the \mathbf{A}-identity on A.
(4) composition in \mathbf{A} is the restriction of composition in \mathbf{B} to the morphisms of \mathbf{A}

Adámek et al, Abstract and Concrete Categories(1990)

Point by point, we’ll pick the definition apart. The first part is pretty clear. The collection of objects in a subcategory is contained in the collection of objects in its “parent”. The second criterion says that the set of morphisms from one object A to another object A' inside the littler category \mathbf{A} should be a subset of all the morphisms from the same A to the same A', but inside \mathbf{B}. That is, there are morphisms from A \to A' in \mathbf{B} that won’t live in the subcategory \mathbf{A}.

The third criterion just states that the identity morphisms on objects should match in both categories. The final criterion tells us that composition inside the subcategory \mathbf{A} only “works” on the morphisms inside \mathbf{A}, but is otherwise the same composition as in \mathbf{B}. We just only perform compositions on morphisms in \mathbf{A} when we’re in \mathbf{A}.

We now define a reflection.

Definition (A- reflection)

Let \mathbf{A} be a subcategory of \mathbf{B}, and let B \in \mathrm{Ob}(\mathbf{B}).

(1) An \mathbf{A}- reflection for B is a morphism B \xrightarrow{r} A from B to A \in \mathrm{Ob}(\mathbf{A}) such that for any morphism B \xrightarrow{f} A' from B to A' \in \mathrm{Ob}(\mathbf{A}) there exists a unique f': A \to A' such that f = f' \circ r.
(2) \mathbf{A} is a reflective subcategory for \mathbf{B} if each \mathbf{B}- object has an \mathbf{A}-reflection. 

Currently, this definition seems a bit abstract. The following sections will illustrate concrete examples of reflections to understand this definition better.

Taking a Group to Its Abelian Form

The Category \mathbf{Grp}

For this example, we’ll be working with the category of groups, \mathbf{Grp}. The objects in this category are groups, and the morphisms are group homomorphisms. Some elements of this category are:

  • (\mathbb{R}, +) \xrightarrow{\phi} (\mathbb{R}^{+}, \cdot), \phi(x) = e^{x}. The real numbers under addition and the positive real numbers under multiplication are both groups, so they’re objects in this category. \phi here is a group homomorphism1, so it’s a morphism in \mathbf{Grp}.
  • The dihedral group D_{6}, and the permutation group S_{3} are also objects in this class. Recall that the dihedral group D_{6} is commonly visualized using the symmetries of an equilateral triangle2, but is commonly given using the following presentation: D_{6} = \langle r,s : r^{3} = s^{2} = 1, sr = r^{-1}s\rangle Here, we see that D_{6} is generated by two elements r and s (r is the rotation by 2\pi/3, and s is one of the reflection lines). S_{3} is the set of permutations on 3 elements– all permutations of the integers \{1,2,3\}. Both of these are groups, and both have 6 elements. If we define \phi: \begin{cases} r \to (123) \\ s \to (12)\end{cases} Then \phi is a group homomorphism that takes D_{6} to S_{3}, and is also thus a morphism in \mathbf{Grp}
  • As one last example, let \mathrm{GL}_{2}(\mathbb{R}) be the general linear group of degree 2. This is the group of invertible 2\times 2 matrices with real entries. Then we can create a group homomorphism \phi: D_{6} \to \mathrm{GL}_{2}(\mathbb{R}) by letting \phi(r) = \begin{bmatrix}\cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta)\end{bmatrix} \qquad \phi(s) = \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} so this \phi is also a morphism in \mathbf{Grp}.

The Category \mathbf{Ab}

A subcategory of \mathbf{Grp} is \mathbf{Ab}, the category of abelian groups. Morphisms are again group homomorphisms, but this time we are only looking at morphisms between abelian groups. Some examples:

(1) (\mathbb{R}, +) \xrightarrow{\phi} (\mathbb{R}^{+}, \cdot), \phi(x) = e^{x}. The real numbers under addition, and the positive real numbers under multiplication are abelian groups, so they’re in the subcategory \mathbf{Ab}.

(2) (\mathbb{Z}_{2} \times \mathbb{Z}_{2}, +) and (\mathbb{Z}_{4}, +) are also abelian groups. We can define a group homomorphism \psi: \mathbb{Z}_{2} \times \mathbb{Z}_{2} \to \mathbb{Z}_{4} by \psi: \begin{cases} (0,0) \to 0 \\ (1,0) \to 1\\ (0,1) \to 2 \\ (1,1) \to 3\end{cases}, so \psi is in the collection of morphisms in \mathbf{Ab}. Of course, it’s also in the morphisms of \mathbf{Grp} as well.

(3) D_{6} is not commutative, so it’s not an object in \mathbf{Ab}

(4)\mathrm{GL}_{2} under matrix multiplication is not abelian, because matrix multiplication is not a commutative operation. So neither this group nor the group homomorphism between D_{6} and \mathrm{GL}_{2} are in \mathbf{Ab}.

Creating the Commutator Subgroup

We now discuss the commutator element. The commutator of two elements g,h in a group, denoted [g,h] is given by [g,h] = g^{-1}h^{-1}gh.

Remark: The order matters here.

Let’s work with a nice easy group of matrices. Let G =\left\{I_{2},\begin{bmatrix}0&1\\1&0\end{bmatrix},\begin{bmatrix}0&1\\-1&-1\end{bmatrix},\begin{bmatrix}-1&-1\\0&1\end{bmatrix},\begin{bmatrix}-1&-1\\1&0\end{bmatrix},\begin{bmatrix}1 & 0\\-1&-1\end{bmatrix}\right\} under matrix multiplication. We’ll name these matrices \{I,A,B,C,D,K\} respectively. This group is not commutative, as shown in the group table below:

\cdotIABCDK
IIABCDK
AAICBKD
BBKDAIC
CCDKIAB
DDCIKBA
KKBADCI

 Next, we’re going to form the commutator subgroup, which is the subgroup of G consisting of all commutators of G. Let’s call this subgroup \tilde{G}. The quotient G/\tilde{G} is always an abelian group, so “quotienting out by” \tilde{G} and working with the cosets gives us an “abelian version” of our previously non-abelian group. 

We’ll calculate a few commutator elements to demonstrate how, but will not run through every combination. \begin{aligned}[I,X]&=[X,I]=I\\ [A,B]&=A^{-1}B^{-1}AB=ADAB=D\\ [B,A]&=B^{-1}A^{-1}BA=DABA=B\\ [C,D]&=C^{-1}D^{-1}CD=CBCD=B\\ \vdots\end{aligned}

Continuing with all combinations, we find that there are only three commutator elements: \{I, B, D\}. Thus3 \tilde{G} = \{I,B,D\}. Now, G/\tilde{G} gives us the left cosets of \tilde{G}: \begin{aligned}A\tilde{G}&= C\tilde{G}=K\tilde{G}=\{A,C,K\}\\ B\tilde{G}&= D\tilde{G}=I\tilde{G}=\{I,B,D\}\end{aligned}Thus, the commutator subgroup is G/\tilde{G} = \{A\tilde{G}, \tilde{G}\}. This two-element group is a bit dull, admittedly, but it certainly is abelian, with the identity element as \tilde{G}.

Back to Reflections

Something else we can do with this little two-element group is map it to (\mathbb{Z}_{2}, +) via the homomorphism \phi: G/\tilde{G} \to \mathbb{Z}_{2}, where \phi(\tilde{G}) = 0, and \phi(A\tilde{G}) = 1

What does any of this have to do with reflections? Recall that G \in \mathrm{Ob}(\mathbf{Grp}), but not in \mathrm{Ob}(\mathbf{Ab}). G/\tilde{G} is in \mathrm{Ob}(\mathbf{Ab}), and so is \mathbb{Z}_{2}

A reflection for G into \mathbf{Ab} is a morphism r such that for any morphism \psi:G \to A', A' \in \mathrm{Ob}(\mathbf{Ab}), we can find a morphism \phi completely contained in \mathbf{Ab} such that \psi = \phi \circ r

Let A'= \mathbb{Z}_{2}, and define \psi: G \to \mathbb{Z}_{2} by \psi: \begin{cases}A \to 1 \\ C \to 1 \\ K \to 1 \\ B \to 0\\D \to 0 \\I \to 0\end{cases} This is certainly a homomorphism (\psi(XY) = \psi(X)\psi(Y)). 

What if we could “bounce” this group G off another group H in \mathbf{Ab}, then move from H to \mathbb{Z}_{2}? That might reveal a little more about \psi than just the seemingly contrived definition we put forth. 

Let r: G \to G/\tilde{G} be the canonical map sending each element of G to the appropriate coset. That is r(A) = A\tilde{G}, r(B) = \tilde{G}, etc. Then the image of r is G/\tilde{G}, and \phi as defined above is the unique morphism that will take G/\tilde{G} to \mathbb{Z}_{2} such that \psi = \phi \circ r

One reflection, r, taking G to some object A in \mathbf{Ab}. Grab a morphism \psi from G to somewhere else in \mathbf{Ab} (we picked \mathbb{Z_{2}}). Then we have to be able to find a unique \phi such that \psi decomposes into the composition of r with that \phi. This same r (and thus the same A) should be able to perform this action for any A', any \psi. In our case, the reflection is the canonical map from G to its commutator subgroup. 

Conclusion

Reflections can perform many different actions to take objects in one category to objects in a subcategory. We focused on reflections making things “abelian” in a nice way, which helped reveal some structures that would have otherwise been hidden.

 



Footnotes

  1. actually an isomorphism
  2. There are 6 symmetry elements. Three reflections, with the reflection lines passing through each of the three vertices, two rotations of angle 2kπ/3, k=1,2, and an identity element that does nothing.
  3. Feel free to verify this is a subgroup.
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