Quotients of Algebraic Structures

I recently started a review of Abstract Algebra with primary focus on Groups. This resulted in a deeper dive into the properties of quotients and Normal subgroups. A common theme across mathematics is the need to construct new instances of particular structures from existing. [In this context I am restricting the term structure to refer to Algebraic Structures.] A common construction is the Quotient : Quotient Groups, Quotient Rings, Quotient Modules, etc.

(Aside : A search for ‘algebraic structures’ returned an extremely thorough taxonomy of such structures math.chapman.edu/~jipsen/structures)

Many constructions derive from an equivalence class defined on the base set $X$ of the structure. An equivalence relation on $X$ is a binary relation (subset of $X \times X$) $~$ on $X$. We commonly use notation $a \sim b$ to say that $(a,b) \in\,\sim$.

• $x \sim x, \forall x \in X$
• $x \sim y \implies y \sim x, \forall x,y \in X$
• $x \sim y, y \sim z \implies x \sim z, \forall x,y,z \in X$

The notation $X / \sim$ is commonly used to refer to the set of equivalence classes of $X$ under the equivalence relation, $\sim$. Where an element $[x] \in X/\sim$ is $[x] = \{ y \vert y \in X, y \sim x\}$.

For quotients of algebraic structures we are explicitly working with sets that are equipped with additional structure (a binary operation satisfying certain conditions, an inverse operation, pairs of binary operations with certain consistency conditions, etc). For an arbitrary equivalence class we can construct the quotient of the underlying set Set(X). When though can we pass the additional structures down to structures defined on $X / \sim$ ?

Compatible Equivalence Relations

For example for a binary operation on $X$ we would like to define a binary operation on $X / \sim$ such that $[x] * [y] = [x * y]$.

The common presentations (that I have experienced) of the topic of quotient groups and quotient rings quickly jumps from discussion of properties of the equivalence relation to discussion of Normal Subgroups and Ideals.

I’ve found it convenient to think directly in terms of equivalence relations that are compatible with the algebraic structure. The convenience comes from making it easier to recall (or reconstruct) the properties of Normal Subgroups, Ideals, and the like - and for providing a different perspective when struggling with a given theorem. With hindsight a helpful reference is Quotient Algebra where we find a very general discussion of quotients of algebraic structures.

An example of the compatibility of an equivalence relation $\sim$ with a binary operation $\cdot$ is given by :

If $g \sim h$ and $g’ \sim h’$ then $g \cdot g’ \sim h \cdot h’$.

or that products of equivalent items are equivalent.

Similarly for any n-ary relation, R, of an algebraic structure one requires that if $g_i \sim h_i$ for $i=1$ to $n$ then $R(g_1, .., g_n) \sim R(h_1, .., h_n)$.

Another way to phrase this is to say we have a relation $\sim\, \subset X \times X$ that is simultaneously a sub-algebra of the product algebra $X \times X$. If a relation $\sim$ defines a sub-algebra of $X \times X$ then we find that for any $n-ary$ operation $R$ on $X$ and n-values $[(g1,h1), (g2,h2), (g3,h3), ..] \in\,\sim$ we have that $(R(g_1, g_2, g_3, ..), R(h_1, h_2, h_3, ..)) \in\, \sim$ which is exactly the compatibility condition above.

Binary Operation

Given a set with only a binary operation $\cdot : X \times X \rightarrow X$ we can explore this consistency. For a binary operation the consistency relation is implied by a weaker condition, that the equivalence relation $\sim$ is closed under left and right diagonal action by the binary operation defined as follows,

$n \sim n’ \implies m \cdot n \sim m \cdot n’$ and $n \cdot m \sim n’ \cdot m$.

A quick proof :

For $n \sim n’$ and $m \sim m’$ we have $m \cdot n \sim m \cdot n’$ and $m \cdot n’ \sim m’ \cdot n’$, so $m \cdot n \sim m’ \cdot n’$.

So compatibility is provided by invariance under these diagonal actions. Defining

\begin{equation} \Delta_{\ell}(g) A = {(g \cdot h, g \cdot h’) \vert (h,h’) \in A} \end{equation}

\begin{equation} \Delta_{r}(g) A = {(h \cdot g, h’ \cdot g) \vert (h,h’) \in A} \end{equation}

\begin{equation} \Delta(g)_{\ell} \sim\, = \Delta(g)_r \sim \,=\, \sim \end{equation}

Similar statements hold beyond the binary case - but are less appealing.

For a trinary operation $T(g_1, g_2, g_3)$ there are 3 notions of an action on $X$. Given any element $g_1, g_2$ of $X \times X$ we define the

• left action by $g_1, g_2$ to be $T(g_1, g_2, \cdot) : X \rightarrow X$
• right action $T(\cdot, g_1, g_2) : X \rightarrow X$
• and ‘sandwich’ action $T(g_1, \cdot, g_2) : X \rightarrow X$

Invariance of $\sim$ under the diagonal action of all 3 ensures compatibility. Given $g_1 \sim h_1$, $g_2 \sim h_2$, and $g_3 \sim h_3$ we have

$T(g_1, g_2, g_3) \sim T(h_1, g_2, g_3)$ by right action. $T(h_1, g_2, g_3) \sim T(h_1, g_2, h_3)$ by left action on $g_3 \sim h_3$. $T(h_1, g_2, h_3) \sim T(h_1, h_2, h_3)$ by ‘sandwich’ (apologies for my laziness in not looking for pre-existing terminology). All together we get $T(g_1, g_2, g_3) \sim T(h_1, h_2, h_3)$

Quotient Group

For a group $G$ consider equivalence relations $\sim$ on $G$ that are compatible with the group structure - or in other words a subgroup of $G \times G$ that is also an equivalence relation.

For a group we have two n-ary operations to worry about :

• Associative binary operation : $\cdot : G \times G \rightarrow G$
• Inverse : ${-1} : G \rightarrow G$

We saw that consistency with any binary operation is equivalent to invariance of $\sim$ under diagonal action of the group. Conveniently this diagonal action is also sufficient for compatibility with the inverse operation.

\begin{align} g &\sim h \newline g \cdot g^{-1} &\sim h \cdot g^{-1} \newline h^{-1} \cdot e &\sim h^{-1} \cdot h \cdot g^{-1} \newline h^{-1} &\sim g^{-1} \end{align}

So having an equivalence relation on $G$ that is invariant under left/right diagonal actions of $G$ is sufficient for compatibility.

From the other direction: How little of the properties of an equivalence relation do we need to impose on a subgroup $H$ of $G \times G$ to ensure we have a compatible equivalence relation? Given a subgroup $H$ that is invariant under left and right diagonal actions.

For a generic subgroup $H$ of $G \times G$ - how much does the left and right diagonal action gain us. Suppose $(g,h) \in H$ and $(h,k) \in H$. Then $(g h^{-1}, e)$ is in H from right action and $(g h^{-1} h, e k) = (g,k)$ so $(g, k)$. So $A$ is transitive. We also have that $g ~ h$ => $h^{-1} g ~ e$ => $h^{-1} ~ g^{-1}$ => $h ~ g$.

So we also have any subgroup of $G \times G$ that is invariant on left and right diagonal group action is also an equivalence class. Or any subgroup of $G \times G$ that contains the diagonal image of $G$ in $G \times G$.

This gives three different paths to a group-compatible equivalence relation - equivalence relation that is closed under the diagonal action of the group, a subgroup of $G \times G$ that contains the diagonal image of $G$ - and the familiar…

Looking at the elements that are equivalent to the identity.

\begin{equation} N = { n \vert n \sim e } \end{equation}

We can quickly identify that $N$ is a normal subgroup of $G$ - $g \cdot h \cdot g^{-1} \sim g \cdot g^{-1} = e$.