Groups with roots of unity

16 Nov 2025

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

  1. A group \(G\) along with a function \(\nu : G\to G\) satisfying \(\nu(x)\nu(y) = xy\).

  2. A group \(G\) along with an element \(\xi\in G\) satisfying \(\xi x = x \xi\) and \(\xi^2 = 1\).

  3. A group object in the category of \(\mathbb{Z}/2\mathbb{Z}\)-sets.

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 well-known example of a group with negation (and the example that motivated the article) is the quaternion 8-group \(Q_8\).

The group of units of any ring is a group with negation. (For the record, I require rings to have a multiplicative identity.) If the ring has characteristic 2, we get a trivial group with negation in which . An interesting question (which I will not answer in this article) is whether every group with negation with \(\xi\neq 1\) can be realized as the group of units of some ring. In the last article, I described how, given a commutative ring \(R\) and a group with negation \((G,\xi)\), one can construct an \(R\)-algebra \(R[(G,\xi)]\) by first constructing the group algebra \(R[G]\) and then taking a quotient, forcing basis elements which are negatives of each other in \(G\) to actually be negatives of each other in the algebra. One example is \(\mathbb{R}[(Q_8,-1)]\), which gives the quaternions. If we instead take \(\mathbb{Z}[(Q_8,-1)]\), we get the quaternions with integer coefficients. The group of units of the latter ring is \(Q_8\), which might lead one to believe that any group with negation \((G,\xi)\) can be realized as the group of units of \(\mathbb{Z}[(G,\xi)]\). However, this is false, as demonstrated by the following example:

  • Let \(G = \langle\omega\rangle\) be the cyclic group of order 8 and let \(\xi = \omega^4\). The ring \(\mathbb{Z}[(G,\xi)]\) is additively a free abelian group generated by \(\{1,\omega,\omega^2,\omega^3\}\), with multiplication determined by the identity \(\omega^4=-1\). Note that

    \[(2\omega + 3\omega^2 + 2\omega^3)(2\omega - 3\omega^2 + 2\omega^3) = 1.\]

    So \(2\omega + 3\omega^2 + 2\omega^3\) is a unit which is not a power of \(\omega\). (I found this example with the help of this paper I found online.)

I made a few remarks at the end of the article, in which I suggested two generalizations. 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, and that the algebra construction in the article would generalize to any monoid with negation without modification.* The second generalization

*In the same way, the familiar definition of group algebra applies to all monoids without modification. Given a ring \(R\) and a monoid \(A\), we can construct the “monoid algebra” \(R[A]\). For example, the polynomial algebra over a commutative ring \(R\) is the monoid algebra \(R[\mathbb{N}]\), where \(\mathbb{N}\) denotes the nonnegative integers under addition. More generally, the polynomial algebra in \(n\) variables is the monoid algebra \(R[\mathbb{N}\oplus\cdots\oplus\mathbb{N}]\), where there are \(n\) copies of \(\mathbb{N}\).