Groups with roots of unity

22 Nov 2025

In an article I wrote in September, I defined a structure called a “group with negation”, giving two equivalent definitions:

A group \(G\) along with a function \(\nu : G\to G\) satisfying \(\nu(x)\nu(y) = xy\) for all \(x,y\in G\).
A group \(G\) along with an element \(\xi\in G\) satisfying \(\xi x = x \xi\) (for all \(x\in G\)) and \(\xi^2 = 1\).

The idea is that \(\nu\) is a “negation” operation, and that \(\xi\) is an element that acts like “\(-1\)”. To get from definition 1 to definition 2, we define \(\xi = \nu(1)\), and to get from definition 2 to definition 1, we define \(\nu(x) = \xi x\). One example of a group with a natural negation structure (and the example that motivated the article) is the quaternion 8-group \(Q_8\).

The group of units of any ring can be made into a group with negation by setting \(\xi=-1\). If the ring has characteristic 2 (or 1), then \(-1=1\), so we get a degenerate group with negation in which \(\xi = 1\) and \(\nu\) is the identity function. In other characteristics, the group of units is a nondegenerate group with negation in which \(\xi \neq 1\) and \(\nu\) has no fixed points.

A homomorphism of groups with negation from \((G,\xi_G)\) to \((H,\xi_H)\) is a group homomorphism \(\varphi : G\to H\) such that \(\varphi(\xi_G) = \xi_H\). This is equivalent to requiring that \(\varphi(\nu_G(x)) = \nu_H(\varphi(x))\) for all \(x\in G\).

More on groups of units

Every group with negation can be embedded into the group of units of some ring. Specifically, given a group with negation \((G,\xi)\), let \(R = \mathbb{Z}[G]/(1+\xi)\). (Here, \(\mathbb{Z}[G]\) denotes the group algebra.) This just the algebra construction I described in the last article, but with \(\mathbb{Z}\) instead of a field as the base ring. There is a natural group homomorphism \(\varphi : G\to R^\times\), namely the composition of the natural embedding \(G\to \mathbb{Z}[G]^\times\) with the quotient map \(\mathbb{Z}[G]^\times\to R^\times\). Note that \(\varphi(\xi) = -1\in R^\times\), so \(\varphi\) is a homomorphism of groups with negation. Moreover, \(\varphi\) is injective, which is perhaps best seen by splitting into cases.

If \(\xi = 1\), then \((1+\xi)\subseteq \mathbb{Z}[G]\) is just the ideal generated by 2, which contains precisely those elements of \(\mathbb{Z}[G]\) all of whose coefficients are even. So \(R \cong \mathbb{F}_2[G]\), and \(\varphi\) is equivalent to the embedding \(G\to\mathbb{F}_2[G]^\times\).

This embedding is always injective, but can fail to be surjective. For example, if \(G =\langle a\rangle\) is the cyclic group of order 5, then the element \(1 + a^2 + a^3\in\mathbb{F}_2[G]\) is a unit, because
\[(1+a^2+a^3)(1+a+a^4) = 1.\]
(Source for this example.)
If \(\xi\neq 1\) (hence \(\xi x\neq x\) for all \(x\in G\)), then the ideal \((1 + \xi)\subseteq\mathbb{Z}[G]\) can be characterized as the set of elements \(\alpha\in\mathbb{Z}[G]\) such that, for all \(x\in G\), the \(\xi x\) component of \(\alpha\) is the equal to the \(x\) component of \(\alpha\). So \(R\) is a free abelian group under addition,* and has characteristic 0. If \(\lvert G\rvert\) is finite, then the dimension of \(R\) is \(\lvert G\rvert/2\); in general, we can get a basis for \(R\) by choosing an element of each equivalence class \(\{x,\xi x\}\subseteq G\). Once we do so, the image of \(\varphi\) consists of the basis elements and their negatives.

Although \(\varphi : G\to R^\times\) is injective, it is (as in the previous case) not necessarily surjective. For example, let \(G = \langle a\rangle\) be the cyclic group of order 8, with \(\xi = a^4\). Then a basis for \(R\) is given by \(\{1,a,a^2,a^3\}\), with multiplication determined by the relation \(a^4 = -1\). Note that
\[(2a + 3a^2 + 2a^3)(2a - 3a^2 + 2a^3) = 1.\]
So \(2a + 3a^2 + 2a^3\) is a unit which is not a power of \(a\). (I found this example with the help of this paper I found online.)

I am pretty sure that the construction of \(R\) from \((G,\xi)\) in this section is left adjoint to the “group of units” functor from the category of rings to the category of groups with negation.

[*To clarify, if we take \(\mathbb{Z}\oplus\mathbb{Z}\) and quotient by the subgroup \(H = \{(n,n)\mid n\in\mathbb{Z}\}\), the result is isomorphic to \(\mathbb{Z}\). Each element of \((\mathbb{Z}\oplus\mathbb{Z})/H\) has a unique representative of the form \((k,0)\) with \(k\in\mathbb{Z}\), or alternatively a unique representative of the form \((0,k)\) with \(k\in\mathbb{Z}\); these two choices of isomorphism are negatives of each other. If we take a direct sum \(\bigoplus_{i\in I}(\mathbb{Z}\oplus\mathbb{Z})\) and quotient by \(\bigoplus_{i\in I}H\), the result is isomorphic to \(\bigoplus_{i\in I}(\mathbb{Z}\oplus\mathbb{Z})/H \cong \bigoplus_{i\in I}\mathbb{Z}\). This is why \(R\) is a free abelian group.]

Generalizations of “group with negation”

I made a few remarks at the end of the previous article, in which I suggested two generalizations of the concept of “group with negation”. The first is to consider monoids rather than groups, obtaining the notion of a “monoid with negation”. This is a pretty natural generalization to consider, given that every ring is a monoid with negation (but only one ring is a group with negation). The second generalization is to replace the exponent in \(\xi^2 = 1\) with an arbitrary positive integer \(n\), obtaining the notion of a “group with \(n\)th root of unity”. Of course, there is a simultaneous generalization, namely a “monoid with \(n\)th root of unity”. The following four definitions of this concept are equivalent:

A monoid \(A\) along with a function \(\nu : A\to A\) satisfying \(\nu(x_1)\cdots\nu(x_n) = x_1\cdots x_n\) for all \(x_1,\cdots,x_n\in A\). (Note that \(n\) is fixed.)
A monoid \(A\) along with an element \(\xi\in A\) satisfying \(\xi x = x\xi\) (for all \(x \in A\)) and \(\xi^n = 1\).
A monoid \(A\) along with a homomorphism (of monoids) \(\psi : \mathbb{Z}/n\mathbb{Z}\to A\) such that the image of \(\psi\) is contained in the center of \(A\).
A monoid object in the category of \(\mathbb{Z}/n\mathbb{Z}\)-sets.

The equivalence between definitions 2 and 3 should be pretty clear: to get from 2 to 3, we define \(\psi(k) = \xi^k\), and to get from 3 to 2, we define \(\xi = \psi(1)\). I will prove the equivalence between definitions 3 and 4 later on in a more general context.

To get from definition 1 to definition 2, we take \(\xi = \nu(1)\). The equation \(\xi^n = 1\) follows from the definitional property of \(\nu\) with \(x_1,\ldots,x_n = 1\). Now let \(x\) be any element of \(A\). Applying the definitional property of \(\nu\) with \(x_1=x\) and \(x_i=1\) for all \(i\neq 1\), we obtain \(\nu(x)\xi^{n-1} = x\). Multiplying by \(\xi\) on the right on both sides, we get \(\nu(x) = x\xi\). In a similar way, we can also show that \(\nu(x) = \xi x\). Hence \(\xi x = x\xi\).

Conversely, to get from definition 2 to definition 1, we define \(\nu(x)=\xi x\). Then

\[\nu(x_1)\cdots\nu(x_n) = \xi x_1\cdots \xi x_n = \xi^n x_1\cdots x_n = x_1\cdots x_n.\]

Note that, if \(n = 1\), a monoid with \(n\)th root of unity is just a monoid. In definition 1, \(\nu\) is forced to be the identity function. In definition 2, \(\xi\) is forced to be 1. In definition 3, \(\psi\) is forced to be the trivial homomorphism (since \(\mathbb{Z}/1\mathbb{Z}\) is the trivial group). And the category of \(\mathbb{Z}/1\mathbb{Z}\)-sets is just the category of sets.

Each of the four definitions comes with a natural notion of homomorphism. (For example, definition 2 suggests that a homomorphism of monoids with \(n\)th root of unity from \((A,\xi_A)\) to \((B,\xi_B\)) should be a monoid homomorphism \(f : A\to B\) that also satisfies \(f(\xi_A) = \xi_B\).) These are all equivalent.

Relationship between groups with \(n\)th root of unity and groups

Let’s walk back the generalizations for a moment and consider groups with \(n\)th root of unity. These are defined by taking definition 1, 2, or 3 in the previous section and replacing the word “monoid” with “group”. (At least in this section, I will use definition 2, and will denote a group with \(n\)th root of unity as a pair \((G,\xi)\).) Groups with \(n\)th root of unity form a category, which I will call \(\mathrm{Gp}_n\). Similarly, I will denote the category of groups as \(\mathrm{Gp}\).

There is, of course, a functor \(U:\mathrm{Gp}_n\to\mathrm{Gp}\) taking a group with \(n\)th root of unity to its underlying group. In the other direction, there is a functor \(D : \mathrm{Gp}\to\mathrm{Gp}_n\) taking a group \(G\) to the degenerate group with \(n\)th root of unity \((G,1)\).

Another way of constructing a group with \(n\) root of unity from a group is as follows: given a group \(G\), let \(F(G) = (G\times\mathbb{Z}/n\mathbb{Z}, (1,1))\). Note that \(1\) denotes the identity element of \(G\) but the generator of \(\mathbb{Z}/n\mathbb{Z}\). The element \((1,1)\in G\times\mathbb{Z}/n\mathbb{Z}\) commutes with everything and has order \(n\), so \(F(G)\) is a well-defined group with \(n\)th root of unity. Moreover, given a group homomorphism \(\varphi : G\to H\), we can define a homomorphism in \(\mathrm{Gp}_n\) from \(F(G)\) to \(F(H)\) by taking \((g,k)\) to \((\varphi(g),k)\). This makes \(F\) into a functor.

Lastly, given a group with \(n\)th root of unity \((G,\xi)\), we can form the quotient group \(G/\langle\xi\rangle\). (Note that \(\langle\xi\rangle\) is normal because it is contained in the center of \(G\).) If \(\varphi : (G,\xi_G)\to (H,\xi_H)\) is a homomorphism in \(\mathrm{Gp}_n\), then \(\varphi\) maps any power of \(\xi_G\) to a power of \(\xi_H\), so \(\varphi\) descends to a group homomorphism \(G/\langle\xi_G\rangle\to H/\langle\xi_H\rangle\). This gives a functor \(Q : \mathrm{Gp}_n\to \mathrm{Gp}\). (The \(Q\) is for “quotient”.)

At this point, we have functors \(U,Q : \mathrm{Gp}_n\to\mathrm{Gp}\) as well as \(F,D : \mathrm{Gp}\to\mathrm{Gp}_n\). These four functors form two adjoint pairs. Specifically, \(F\) is left adjoint to \(U\), and \(Q\) is left adjoint to \(D\). To clarify:

Let \(G\) be a group and let \((H,\xi)\) a group with \(n\)th root of unity. A homomorphism \(F(G)\to(H,\xi)\) is equivalently a group homomorphism \(\varphi : G\times\mathbb{Z}/n\mathbb{Z}\to H\) such that \(\varphi(1,1) = \xi\). From such a homomorphism, we can get a group homomorphism \(\tilde{\varphi} : G\to H\) by defining \(\tilde{\varphi}(g) = \varphi(g,0)\). Conversely, given a group homomorphism \(\chi : G\to H\), we can get a group homomorphism \(\hat{\chi} : G\times\mathbb{Z}/n\mathbb{Z}\to H\) by defining \(\hat{\chi}(g,k) = \chi(g)\xi^k\). Note that \(\hat{\chi}(1, 1) = \xi\), so \(\hat{\chi}\) is indeed a homomorphism in \(\mathrm{Gp}_n\) from \(F(G)\) to \((H, \xi)\). It is not hard to check that these operations (i.e. the ones that I am denoting with a squiggle and a hat) are inverse to each other.

The intuition here is that a homomorphism from \(F(G)\) to \((H,\xi)\) is forced in how it acts on the \(\mathbb{Z}/n\mathbb{Z}\) component, but can act freely on the \(G\) component.
Let \((G,\xi)\) be a group with \(n\)th root of unity and let \(H\) be a group. There is a natural correspondence between homomorphisms from \(G/\langle\xi\rangle\) to \(H\) and homomorphisms from \(G\) to \(H\) that send \(\xi\) to 1. In other words, a homomorphism in \(\mathrm{Gp}\) from \(Q(G,\xi)\) to \(H\) is the same as a homomorphism in \(\mathrm{Gp}_n\) from \((G,\xi)\) to \(D(H)\).

As I will show later, these functors and the adjunctions between them are special cases of a more general construction. Specifically, \(U\) and \(D\) (the right adjoints) are both special cases of “restriction of scalars”, and \(F\) and \(Q\) (the left adjoints) are both special cases of “extension of scalars”.

Algebra construction

I will call a monoid with \(n\)th root of unity \((A,\xi)\) “primitive” if the order of \(\xi\) is \(n\). (It is always some divisor of \(n\).) This is equivalent to requiring that the structure homomorphism \(\psi : \mathbb{Z}/n\mathbb{Z}\to A\) be injective. Instead of “primitive monoid with \(n\)th root of unity”, I will usually say “monoid with primitive \(n\)th root of unity”, as the latter mirrors the familiar term “primitive \(n\)th root of unity”. Given any monoid \(A\) along with a central element \(\xi\in A\) of finite order, there is a unique positive integer \(n\) such that \((A,\xi)\) is a monoid with primitive \(n\)th root of unity. (Namely \(n = \mathrm{ord}(\xi)\).)

If \((G,\xi)\) is a group with primitive \(n\)th root of unity, then \(G\) is divided into equivalence classes of size \(n\), namely the cosets of \(\langle\xi\rangle\). We can construct a \(\mathbb{C}\)-algebra \(\mathbb{C}[(G,\xi)]\) by taking the group algebra \(\mathbb{C}[G]\) and then quotienting by the ideal \(I\) generated by \(\xi - e^{i\tau / n}\). (Yes, I am using \(\tau\) to mean \(2\pi\).) This causes \(\xi^k\) (or more formally, the basis element of \(\mathbb{C}[G]\) corresponding to \(\xi^k\)) to be identified with \(e^{i\tau k/n}\) for all \(k\). More generally, for all \(g\in G\) and \(k\in\mathbb{Z}/n\mathbb{Z}\), the basis element \(\xi^k g\) is identified with \(e^{i\tau k/n}g\).

(If the order of \(\xi\in G\) were less than \(n\), say \(k < n\), then the elements \(1, e^{i\tau k/n}\in\mathbb{C}[G]\) would be identified, causing everything to collapse to 0. This is why I require that \((G,\xi)\) is a group with primitive \(n\)th root of unity.)

Fix a complete set of \(\langle\xi\rangle\)-coset representatives \(\{a_j\}_{j\in J}\). If \(G\) is finite, then we will have \(\vert J\vert = \vert G\vert/n\). Let \(V\) be the \(\mathbb{C}\)-vector space freely spanned by elements denoted \(\{v_j\}_{j\in J}\). We construct a linear map \(L : \mathbb{C}[G]\to V\) as follows. Given any element \(g\in G\), there is a unique \(j\in J\) and \(k\in\mathbb{Z}/n\mathbb{Z}\) such that \(g = \xi^ka_j\). Define \(L(g)=e^{i\tau k/n}v_j\). \(L\) is clearly surjective, because for all \(j\in J\), we have \(L(a_j) = v_j\). And it is not too hard to prove that the kernel of \(L\) is precisely \(I\).*

So, by the first isomorphism theorem for vector spaces, we have a linear isomorphism between \(\mathbb{C}[(G,\xi)]\) and \(V\). This gives an idea of the size of \(\mathbb{C}[(G,\xi)]\): it has one basis vector for each coset of \(\langle\xi\rangle\) in \(G\) (although there is not a canonical basis). In particular, if \(G\) is finite, then \(\mathbb{C}[(G,\xi)]\) has dimension \(\vert G\vert/n\).

[*Note that \(\xi - e^{i\tau /n}\) is in the center of \(\mathbb{C}[G]\), so \(I\) is the set of elements of the form \(\alpha(\xi - e^{i\tau / n})\) with \(\alpha\in\mathbb{C}[G]\). The \(I\subseteq\ker L\) direction is pretty straightforward. The \(\ker L\subseteq I\) direction comes down to the fact that, if \(w\in \mathbb{C}^n\) is the vector \((1, e^{i\tau/n}, \ldots, e^{i\tau(n-1)/n})\), then the “orthogonal complement” of \(w\) — defined using the dot product, not the inner product — has a basis given by all \(n-1\) shifted versions of \((-e^{i\tau/n}, 1, 0, 0, \ldots)\).]

\(G\)-monoids and \(G\)-groups

I have claimed that a monoid with \(n\)th root of unity is equivalently a monoid object in the category of \(\mathbb{Z}/n\mathbb{Z}\)-sets, but have so far not elaborated on this.

Let \(G\) be an abelian group (keeping in mind the special case \(G=\mathbb{Z}/n\mathbb{Z}\)), and let \(G\mathrm{Set}\) denote the category of \(G\)-sets (meaning sets equipped with an action of \(G\)). A morphism from \(X\) to \(Y\) in this category is given by a function \(\varphi : X\to Y\) such that \(\varphi(gx) = g\varphi(x)\) for all \(g\in G\) and \(x\in X\).

Given two objects \(X,Y\in G\mathrm{Set}\), the set of morphisms \(\mathrm{Hom}(X,Y)\) can itself be made into a \(G\)-set, by defining (for all \(g\in G\) and \(\varphi\in\mathrm{Hom}(X,Y)\)) \((g\varphi)(x) = g\varphi(x)\). Note that \(g\varphi\) is again a morphism because for any \(h\in G\) and \(x\in X\),

\[(g\varphi)(hx) = g\varphi(hx) = gh\varphi(x) = hg\varphi(x) = h(g\varphi)(x).\]

This uses the fact that \(G\) is abelian.

Another way of constructing a \(G\)-set out of two \(G\)-sets is as follows. If \(X\) and \(Y\) are two \(G\)-sets, then \(G\) can act on the set \(X\times Y\) by \(g(x,y) = (gx,g^{-1}y)\). (The fact that this is an action relies on the fact that \((gh)^{-1} = g^{-1}h^{-1}\), which is only true because \(G\) is abelian.) Denote the set of orbits as \(X\otimes Y\), and denote the orbit of \((x,y)\in X\times Y\) as \(x\otimes y\). For all \(x\in X\), \(y\in Y\), and \(g\in G\), we have

\[x\otimes gy = gx\otimes g^{-1}gy = gx\otimes y.\]

We can make \(X\otimes Y\) into a \(G\)-set by defining \(g(x\otimes y) = gx\otimes y\). To show that this is well-defined, we must show that \(ghx\otimes h^{-1}y = gx\otimes y\) for all \(h\in G\). Indeed,

\[ghx\otimes h^{-1}y = hgx\otimes h^{-1}y = gx\otimes y.\]

Some relations satisfied by \(\mathrm{Hom}\) and \(\otimes\) are

\(X\otimes Y \cong Y\otimes X\). The isomorphism sends \(x\otimes y\in X\otimes Y\) to \(y\otimes x\).
\((X\otimes Y) \otimes Z \cong X\otimes (Y\otimes Z)\). The isomorphism sends \((x\otimes y)\otimes z\) to \(x\otimes(y\otimes z)\).
\(G\otimes X \cong X\). (Here, \(G\) is acting on itself by multiplication.) The isomorphism from \(X\) to \(G\otimes X\) sends \(x\in X\) to \(1\otimes x\), and its inverse sends \(g\otimes x\in G\otimes X\) to \(gx\).
\(\mathrm{Hom}(G,X) \cong X\). The isomorphism from \(\mathrm{Hom}(G,X)\) to \(X\) sends \(f\) to \(f(1)\), and its inverse sends \(x\in X\) to the morphism \(g\mapsto gx\).
\(\mathrm{Hom}(X,\mathrm{Hom}(Y,Z)) \cong \mathrm{Hom}(X\otimes Y, Z)\).

To see why relation 5 holds, note that a morphism \(\varphi : X\to \mathrm{Hom}(Y,Z)\) gives (by “uncurrying”) a function \(f : X\times Y\to Z\). The fact that \(\varphi\) is morphism of \(G\)-sets translates to the identity \(f(gx,y) = gf(x,y)\), and the fact that \(\varphi(x)\) is a morphism of \(G\)-sets for all \(x\) translates to the identity \(f(x,gy) = gf(x,y)\). Similarly, a morphism \(\varphi : X\otimes Y\to Z\) gives a function \(f : X\times Y \to Z\), namely \(f(x,y) = \varphi(x\otimes y)\). The fact that \(f\) factors through \(X\otimes Y\) means that \(f(gx,y) = f(x,gy)\), and the fact that \(\varphi\) is a morphism means that \(f(gx, y) = gf(x,y)\). So \(\mathrm{Hom}(X,\mathrm{Hom}(Y,Z))\) and \(\mathrm{Hom}(X\otimes Y, Z)\) are both in bijection with the set of functions \(f : X\times Y\to Z\) such that

\[f(gx,y) = f(x,gy) = gf(x,y)\]

for all \(g\in G\), \(x\in X\), and \(y\in Y\). The fact that this is an isomorphism of \(G\)-sets and not just a bijection comes down to the fact that the action of \(G\) on Hom-objects is defined pointwise.

Perhaps I should also mention that \(\otimes\) is functorial: If we have two morphisms \(\alpha : X\to X'\) and \(\beta : Y\to Y'\) in \(G\mathrm{Set}\), there is a morphism \(\alpha\otimes\beta : X\otimes Y\to X'\otimes Y'\) sending \(x\otimes y\) to \(\alpha(x)\otimes\beta(y)\). Similarly, if we have two morphisms \(\alpha : X'\to X\) and \(\beta : Y\to Y'\), there is a morphism \(\mathrm{Hom}(\alpha,\beta) : \mathrm{Hom}(X,Y)\to\mathrm{Hom}(X',Y')\) sending \(\varphi\) to \(\beta\circ\varphi\circ\alpha\). (Note the contravariance in the first argument.)

Since \(G\mathrm{Set}\) is a monoidal category with \(\otimes\) as the monoidal operation and \(G\) as the identity object, it has a notion of “monoid object”, which I will call a \(G\)-monoid. This is an object \(A\in G\mathrm{Set}\) along with morphisms \(\eta : G\to A\) and \(\mu : A\otimes A\to A\) satisfying the following identity and associativity laws:

\(A\cong G\otimes A\xrightarrow{\eta\otimes\mathrm{id}} A\otimes A\xrightarrow{\mu}A\) is the identity map on \(A\).
\(A\cong A\otimes G\xrightarrow{\mathrm{id}\otimes\eta} A\otimes A\xrightarrow{\mu}A\) is the identity map on \(A\).
\((A\otimes A)\otimes A\xrightarrow{\mu\otimes\mathrm{id}} A\otimes A\xrightarrow{\mu} A\) is the same as \((A\otimes A)\otimes A\cong A\otimes(A\otimes A) \xrightarrow{\mathrm{id}\otimes\mu} A\otimes A\xrightarrow{\mu} A\).

To convert this definition into a more concrete form, recall that a morphism \(\eta : G\to A\) is equivalently an element \(\hat{\eta}\in A\), specifically \(\hat{\eta} = \eta(1)\). And a morphism \(\mu : A\otimes A\to A\) is equivalently a function \(\hat{\mu} : A\times A\to A\) satisfying the identity \(\hat{\mu}(gx, y) = \hat{\mu}(x,gy) = g\hat{\mu}(x,y)\). Specifically, \(\hat{\mu}(x,y) = \mu(x\otimes y)\). We can now translate the laws 1–3 into terms of \(\hat{\eta}\) and \(\hat{\mu}\).

If we apply the morphism in law 1 to an element \(x\in A\), we get \(x\mapsto 1\otimes x\mapsto \hat{\eta}\otimes x\mapsto \hat{\mu}(\hat{\eta}, x)\). So law 1 states that \(\hat{\mu}(\hat{\eta},x) = x\).
If we apply the morphism in law 2 to an element \(x\in A\), we get \(x\mapsto x\otimes 1\mapsto x\otimes\hat{\eta}\mapsto \hat{\mu}(x, \hat{\eta})\). So law 2 states that \(\hat{\mu}(x, \hat{\eta}) = x\).
If we apply the two morphisms in law 3 to an element \((x\otimes y)\otimes z\in(A\otimes A)\otimes A\), we get \(\hat{\mu}(\hat{\mu}(x,y),z)\) respectively \(\hat{\mu}(x,\hat{\mu}(y,z))\). So law 3 states that \(\hat{\mu}(\hat{\mu}(x,y),z) = \hat{\mu}(x,\hat{\mu}(y,z))\).

In other words, laws 1–3 simply state that \(A\) is a monoid with \(\hat{\eta}\) as the identity element and \(\hat{\mu}\) as the multiplication. So we arrive at the following equivalent definition of a \(G\)-monoid: A \(G\)-monoid is a \(G\)-set \(A\) that is also a monoid, subject to the compatibility condition

\[gx\cdot y = x\cdot gy = g(x\cdot y).\tag{1}\]

for all \(g\in G\) and \(x,y\in A\). (Here \(\cdot\) denotes the monoid multiplication.)

I claim that this is the same as a monoid \(A\) along with a homomorphism of monoids \(G\to A\) whose image is contained in the center of \(A\). To prove this, let \(A\) be a monoid.

Suppose that \(A\) is a also a \(G\)-set, and satifies the identity (1). Define a monoid homomorphism \(\varphi : G\to A\) by \(\varphi(g) = g1_A\). This is a homomorphism because \(\varphi(1_G) = 1_G1_A = 1_A\) and for all \(g,h\in G\),
\[\begin{gather*} \varphi(gh) = gh1_A = gh(1_A\cdot 1_A) = g(1_A\cdot h1_A) \\ = g1_A\cdot h1_A = \varphi(g)\cdot\varphi(h). \end{gather*}\]
And the image of \(\varphi\) is contained in the center of \(A\), because for all \(g\in G\) and \(x\in A\),
\[\begin{gather*} \varphi(g)\cdot x = g1_A\cdot x = g(1_A\cdot x) = g(x\cdot 1_A) = x\cdot g1_A = x\cdot\varphi(g) \end{gather*}\]
Conversely, suppose that we have a monoid homomorphism \(\varphi : G \to A\) such that the image of \(\varphi\) is contained in the center of \(A\). Let \(G\) act on \(A\) by \(gx = \varphi(g)\cdot x\). Then for all \(g\in G\) and \(x,y\in A\), the expressions \(gx\cdot y\), \(x\cdot gy\), and \(g(x\cdot y)\) are all equal to \(\varphi(g)\cdot x\cdot y\). So the identity (1) holds.
These conversions are inverse to each other: If we start with the \(G\)-action and define \(\varphi(g) = g1_A\), then \(gx = g(1_A\cdot x) = g1_A\cdot x = \varphi(g)\cdot x\). And if we start with the homomorphism \(\varphi : G\to A\) and define \(gx = \varphi(g)\cdot x\), then \(\varphi(g) = \varphi(g)\cdot 1_A = g1_A\).

In the special case \(G = \mathbb{Z}/n\mathbb{Z}\), this proves the equivalence of definitions 3 and 4 of “monoid with \(n\)th root of unity.”

We have generalized the concept of a monoid with \(n\)th root of unity to that of a \(G\)-monoid, where \(G\) is any abelian group. We can similarly generalize the concept of a group with \(n\)th root of unity by a defining a “\(G\)-group” to be a \(G\)-monoid that happens to be a group. Then a \(\mathbb{Z}/n\mathbb{Z}\)-group is a \(\mathbb{Z}/n\mathbb{Z}\)-monoid that happens to be a group, i.e. a monoid with \(n\)th root of unity that happens to be a group, i.e. a group with \(n\)th root of unity. A \(G\)-group can be characterized as a group \(H\) along with a group homomorphism \(G \to H\) whose image is contained in the center of \(H\). It can also be characterized as a group \(H\) that is simultaneously a \(G\)-set and satisfies identity (1) for all \(g\in G\) and \(x,y\in H\).

A morphism of \(G\)-monoids \(\varphi : A\to B\) is a function that is both a monoid homomorphism and a morphism of \(G\)-sets. If \(\psi_A : G\to A\) and \(\psi_B : G\to B\) are the structure homomorphisms, then the condition that \(\varphi\) be a morphism of \(G\)-sets translates to the identity \(\varphi(\psi_A(g)\cdot x) = \psi_B(g)\cdot\varphi(x)\) for all \(g\in G\) and \(x\in A\). Taking \(x = 1\), this implies that \(\varphi(\psi_A(g)) = \psi_B(g)\) for all \(g\in G\). Conversely, if \(\varphi : A\to B\) is a monoid homomorphism satisfying the latter equation, then for all \(g\in G\) and \(x\in A\), we have

\[\varphi(\psi_A(g)\cdot x) = \varphi(\psi_A(g))\cdot\varphi(x) = \psi_B(g)\cdot x.\]

So a morphism of \(G\)-monoids \(A\to B\) can also be characterized as a monoid homomorphism \(\varphi : A\to B\) such that \(\varphi\circ\psi_A = \psi_B\).* I will denote the category of \(G\)-monoids as \(G\mathrm{Mon}\). I will similarly denote the category of \(G\)-groups as \(G\mathrm{Gp}\); this is a full subcategory of \(G\mathrm{Mon}\), since in general any monoid homomorphism between groups is a group homomorphism.

Surely you have noticed the similarities between \(G\)-sets and modules over a commutative ring, and between \(G\)-monoids and algebras over a commutative ring.

[*Normally, to check that a function \(\varphi : A\to B\) is a monoid homomorphism, we need to check that \(\varphi(1) = 1\). But in this case, we need not check that separately, since it follows from \(\varphi(\psi_A(1)) = \psi_B(1)\).]

Restriction and extension of scalars

In this section, I will generalize the constructions discussed in the section “Relationship between groups with \(n\)th root of unity and groups”.

Suppose we have a homomorphism of abelian groups \(\alpha : G\to H\). Given any \(H\)-monoid \(A\) with structure homomorphism \(\psi_A : H\to A\), we get a \(G\)-monoid \(\alpha^* A\), which has the same underlying monoid but has the structure homomorphism \(\psi_A\circ\alpha : G\to A\). (Since the image of \(\psi_A\) is contained in the center of \(A\), the image of \(\psi_A\circ\alpha\) is certainly contained in the center of \(A\).) Any morphism of \(H\)-monoids \(\varphi : A\to B\) is automatically a morphism of \(G\)-monoids from \(\alpha^* A\) to \(\alpha^* B\): since \(\varphi\circ\psi_A = \psi_B\), it follows that \(\varphi\circ\psi_A\circ\alpha = \varphi\circ\psi_B\circ\alpha\). So we have a functor \(\alpha^* : H\mathrm{Mon}\to G\mathrm{Mon}\). We can call this “restriction of scalars” by analogy with algebras over fields.

If \(A\) is an \(H\)-group, then \(\alpha^* A\) is a \(G\)-group, so \(\alpha^*\) restricts to a functor \(H\mathrm{Gp}\to G\mathrm{Gp}\).

On the other hand, given a \(G\)-monoid \(A\), we can construct an \(H\)-monoid \(\alpha_* A\) as follows. If we view \(H\) as a \(G\)-group with structure homomorphism \(\alpha\), then we can take the tensor product of \(G\)-sets \(H\otimes A\). We can make \(H\otimes A\) into a monoid by defining

\[(h\otimes a)(h'\otimes a') = hh'\otimes aa'\]

for all \(h,h'\in H\) and \(a,a'\in A\). This is well-defined because for any \(g,g'\in G\), we have

\[\begin{gather*} (gh\otimes g^{-1}a)(g'h'\otimes g'^{-1}a') = (gh)(g'h')\otimes (g^{-1}a)(g'^{-1}a') \\ = gg'(hh')\otimes g^{-1}g'^{-1}(aa') = hh'\otimes aa'. \end{gather*}\]

Let \(\alpha_* A = H\otimes A\), with homomorphism \(\psi : H\to H\otimes A\) given by \(\psi(h) = h\otimes 1\). (Note that the image of \(\psi\) is indeed in the center of \(H\otimes A\).) The corresponding action of \(H\) on \(H\otimes A\) is given by

\[h'(h\otimes a) = \psi(h')(h\otimes a) = (h'\otimes 1)(h\otimes a) = h'h\otimes a.\]

Given any morphism of \(G\)-monoids \(\varphi : A\to B\), we get a morphism of \(H\)-monoids \(\alpha_*\varphi : \alpha_*A\to\alpha_* B\) by defining \(\alpha_*\varphi(h\otimes a) = h\otimes\varphi(a)\). We already know that you can “tensor” two morphisms of \(G\)-sets (in this case, the identity map on \(H\) along with \(\varphi\)) to get a morphism of \(G\)-sets, so \(\alpha_*\varphi\) is well-defined. And it is straightforward to show that \(\alpha_*\varphi\) is a morphism of \(H\)-sets as well as a monoid homomorphism, so \(\alpha_*\varphi\) is indeed a morphism of \(H\)-monoids. So we have a functor \(\alpha_* : G\mathrm{Mon}\to H\mathrm{Mon}\) (the identity and associativity laws are pretty obvious). We can call this “extension of scalars”.

If \(A\) is a \(G\)-group, then \(\alpha_*A\) is an \(H\)-group: The inverse of \(h\otimes a\in\alpha_*A\) is given by \(h^{-1}\otimes a^{-1}\). So \(\alpha_*\) restricts to a functor \(G\mathrm{Gp}\to H\mathrm{Gp}\).

I claim that \(\alpha_*\) is left adjoint to \(\alpha^*\). To see this, let \(A\) be a \(G\)-monoid and let \(B\) be an \(H\)-monoid, with structure homomorphisms \(\psi_A : G\to A\) and \(\psi_B : H\to B\). A morphism in \(H\mathrm{Mon}\) from \(\alpha_* A\) to \(B\) is a function \(\varphi : H\otimes A\to B\) such that:

\(\varphi(hh'\otimes aa') = \varphi(h\otimes a)\varphi(h'\otimes a')\) for all \(h,h'\in H\) and \(a,a'\in A\).
\(\varphi(h\otimes 1) = \psi_B(h)\) for all \(h\in H\).

This is equivalent to a function \(\varphi : H\times A\to B\) such that:

\(\varphi(\alpha(g)h, a) = \varphi(h, \psi_A(g)a)\) for all \(g\in G\), \(h\in H\), and \(a\in A\). (So \(\varphi\) factors through \(H\otimes A\).)
\(\varphi(hh', aa') = \varphi(h, a)\varphi(h', a')\) for all \(h,h'\in H\) and \(a,a'\in A\).
\(\varphi(h, 1) = \psi_B(h)\) for all \(h\in H\).

On the other hand, a morphism in \(G\mathrm{Mon}\) from \(A\) to \(\alpha^* B\) is a function \(\chi : A\to B\) such that:

\(\chi(aa') = \chi(a)\chi(a')\) for all \(a,a'\in A\).
\(\chi(\psi_A(g)) = \psi_B(\alpha(g))\) for all \(g\in G\).

To get from \(\varphi\) to \(\chi\), we define \(\chi(a) = \varphi(1, a)\). Property 1 of \(\chi\) follows from property 2 of \(\varphi\) with \(h = h' = 1\). Property 2 of \(\chi\) follows from properties 1 and 3 of \(\varphi\):

\[\chi(\psi_A(g)) = \varphi(1, \psi_A(g)) = \varphi(\alpha(g), 1) = \psi_B(\alpha(g)).\]

To get from \(\chi\) to \(\varphi\), we define \(\varphi(h, a) = \psi_B(h)\chi(a)\). To prove property 1 of \(\varphi\), we need both properties of \(\chi\):

\[\begin{gather*} \varphi(h, \psi_A(g)a) = \psi_B(h)\chi(\psi_A(g)a) = \psi_B(h)\chi(\psi_A(g))\chi(a) = \psi_B(h)\psi_B(\alpha(g))\chi(a) \\ = \psi_B(h\alpha(g))\chi(a) = \psi_B(\alpha(g)h)\chi(a) = \varphi(\alpha(g)h, a). \end{gather*}\]

Property 2 of \(\varphi\) follows from property 1 of \(\chi\) and the fact that the image of \(\psi_B\) is contained in the center of \(B\). For property 3 of \(\varphi\), we need the fact that \(\chi(1) = 1\), which follows from property 2 of \(\chi\) with \(g=1\).

It is not hard to prove that the definition of \(\chi\) in terms of \(\varphi\) and the definition of \(\varphi\) in terms of \(\chi\) are inverses of each other. This pretty much proves the adjunction.

As a special case, fix some positive integer \(n\) and consider the unique homomorphism \(\alpha : 1\to \mathbb{Z}/n\mathbb{Z}\), where \(1\) is the trivial group. The corresponding functor \(\alpha^* : \mathbb{Z}/n\mathbb{Z}\mathrm{Gp}\to \mathrm{Gp}\) is the functor taking a group with \(n\)th root of unity to its underlying group. We called this functor \(U\) in a previous section. We also have a functor \(\alpha_* : \mathrm{Gp}\to \mathbb{Z}/n\mathbb{Z}\mathrm{Gp}\). Unraveling the definition, \(\alpha_*\) takes a group \(A\) to the direct product group \(\mathbb{Z}/n\mathbb{Z}\times A\), with homomorphism \(\psi : \mathbb{Z}/n\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}\times A\) given by \(\psi(k) = (k,1)\). This is the same as the functor \(F\) from a previous section, which we already saw was left adjoint to \(U\).

We can also take \(\alpha\) to be the unique homomorphism \(\mathbb{Z}/n\mathbb{Z}\to 1\). Now \(\alpha^* : \mathrm{Gp}\to\mathbb{Z}/n\mathbb{Z}\mathrm{Gp}\) is the functor (which we called \(D\) earlier) taking a group to the corresponding degenerate group with \(n\)th root of unity. On the other hand, \(\alpha_* : \mathbb{Z}/n\mathbb{Z}\mathrm{Gp}\to\mathrm{Gp}\) takes a group with \(n\)th root of unity \(A\) to the tensor product \(A\otimes 1\). This is the cartesian product \(A\times 1\) (which we can identify with \(A\)) modulo the equivalence relation \((a, 1)\sim(\xi^k a, 1)\) for all \(a\in A\) and \(k\in\mathbb{Z}/n\mathbb{Z}\). So \(\alpha_*\) is the quotient functor \(Q\) from a previous section, which we already saw was left adjoint to \(D\).

There is nothing really special about \(\mathbb{Z}/n\mathbb{Z}\) in these examples. Given any abelian group \(G\), there is a unique homomorphism \(\alpha : 1\to G\) and a unique homomorphism \(\beta : G\to 1\), giving functors \(\alpha^*, \beta_* : G\mathrm{Gp}\to\mathrm{Gp}\) and \(\alpha_*, \beta^* : \mathrm{Gp}\to\mathbb{Z}/n\mathbb{Z}\mathrm{Gp}\). We can also use \(\mathrm{Mon}\) and \(G\mathrm{Mon}\) instead of \(\mathrm{Gp}\) and \(G\mathrm{Gp}\).