Groups with negatives

9 Sep 2025

The group algebra of a group \(G\) over a field \(K\), denoted \(K[G]\), is a vector space spanned by basis vectors that correspond to elements of \(G\). I will write the basis vectors as \([g]\) for \(g \in G\). We can multiply any two vectors in \(K[G]\) using the bilinear operation generated by \([g][h] = [gh]\) for \(g, h \in G\).

The quaternion 8-group \(Q_8\) is defined as the set with 8 elements denoted \(\pm 1\), \(\pm i\), \(\pm j\), \(\pm k\), along with the multiplication operation given in this rather low-resolution image:

Q8 multiplication

We can define the quaternions in terms of the quaternion 8-group by taking the group algebra \(\mathbb{R}[Q_8]\) and then quotienting by the subspace

\[S = \mathrm{span}\{[+1] + [-1], [+i] + [-i], [+j] + [-j], [+k] + [-k]\}.\]

This quotient causes the vector corresponding to \(-i\) to actually be the negative of the vector corresponding to \(+i\), and so on. For the quotient to be an algebra, the subspace \(S\) must be a two-sided ideal; that is, multiplying an element of \(S\) by any element of \(R[Q_8]\) (in either order) must always give an element of \(S\). It is in fact an ideal, the reason being that if you take two elements of \(Q_8\) which are negatives of each other, and multiply them (on the left or on the right) by the same element of \(Q_8\), the results are negatives of each other. Or in other words, \(-(xy) = (-x)y = x(-y)\) for all \(x, y \in Q_8\), where the negation operation on \(Q_8\) is the defined in the obvious way.

We can generalize this construction from \(Q_8\) to any group in which you can “negate” elements. I can think of three equivalent ways to define this concept:

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

  2. A group with negation is a group \(G\) along with a chosen element \(\xi\) such that \(\xi x = x\xi\) for all \(x \in G\) (that is, \(\xi\) is in the center of \(G\)) and \(\xi^2 = 1\).

  3. A group with negation is a group object in the category of \(\mathbb{Z}/2\)-sets (i.e. sets with an action of \(\mathbb{Z}/2\), or equivalently, sets equipped with an involution), where the tensor product of \(\mathbb{Z}/2\)-sets is the “half-product” defined in this old article of mine.

To get from definition 1 to definition 2, we take \(\xi = \nu(1)\). Note that that \(\nu(1)\nu(1) = 1\) and that, more generally, \(\nu(1)\nu(x) = x\) for any \(x \in G\). By multiplying both sides of the latter equation by \(\nu(1)\) on the left, we obtain \(\nu(x) = \nu(1)x\) (implying that \(\nu\) is an bijection, and in fact an involution). By instead taking \(\nu(x)\nu(1) = x\) and multiplying both sides by \(\nu(1)\) on the right, we obtain \(\nu(x) = x\nu(1)\), and hence \(\nu(1)x = x\nu(1)\). So \(\xi = \nu(1)\) has the necessary properties.

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

\[\nu(x)\nu(y) = \xi x \xi y = \xi^2 x y = xy.\]

Of course, if \(\nu(x)\) is defined as \(\xi x\), then \(\nu(1) = \xi\). Similarly, if we start with definition 1 and define \(\xi\) as \(\nu(1)\), then \(\xi x = \nu(1) x = \nu(x)\) for any \(x \in G\). So our methods of switching between the definitions are inverse to each other. We need to check this because being a “group with negation” is not a property that a group can have, it is an additional structure that is given to a group. A group can admit multiple negation structures; for example, the Klein 4-group admits four, because every element is central and squares to 1.

You may notice that, in any group, we can simply define \(\nu(x) = x\), or equivalently, \(\xi = 1\). That is, any group can be made into a rather trivial “group with negation.” You could exclude these trivial examples by requiring that \(\xi \neq 1\), but that would cause groups with negation to no longer be “algebraic structures” in the sense of universal algebra (and would cause definition 3 to no longer hold). Under the current definition, the category of groups with negation (with the morphisms being group homomorphisms that take \(\xi\) to \(\xi\)) is nicer than it would be under the exclusive definition. I can also think of arguments in favor of the exclusive definition, however. For example, in any nontrivial group with negation, \(\nu\) has no fixed points (and therefore has \(\lvert G\rvert/2\) orbits), because if \(\xi x = x\) for any \(x\), then \(\xi = 1\). So there is quite a sharp contrast between trivial and nontrivial examples.

The identity \(\nu(xy) = \nu(x)y = x\nu(y)\) holds in any group with negation. Indeed, it is obvious if we write it as \(\xi xy = \xi xy = x\xi y\). So the following construction is valid.

We define the algebra of a group with negation \((G, \nu)\) over a field \(K\) as

\[K[(G,\nu)] = K[G]/\mathrm{span}\{[x] + [\nu(x)]\}_{x\in G}.\]

If \(\nu\) is the identity, the algebra is 0-dimensional, because every basis vector is forced to be its own negative. Otherwise, it is \(\lvert G \rvert /2\)-dimensional. Some examples over the reals:

  • The cyclic group \(\{1, a\}\) with \(\xi = a\) gives \(\mathbb{R}\).
  • The cyclic group \(\{1, a, a^2, a^3\}\) with \(\xi = a^2\) gives \(\mathbb{C}\).
  • \(Q_8\) with the natural negation \(\xi = -1\) gives \(\mathbb{H}\), the quaternions.

Some additional notes:

  • The identity \(\nu(x)^{-1} = \nu(x^{-1})\) holds in every group with negation.
  • We can also define a “monoid with negation”; definitions 1, 2, and 3 are still equivalent if we replace every occurence of “group” with “monoid”. One difference between monoids with negation and groups with negation is that the theorem “\(\nu\) is either the identity or has no fixed points” no longer holds, because, in a monoid, \(\xi x = x\) for some \(x\) does not imply \(\xi = 1\). For example, the integers under multiplication are a monoid with negation, and \(-0 = 0\).
  • Every algebra is a monoid with negation, so there is a forgetful functor from the category of \(K\)-algebras to the category of monoids with negation. I suspect that the left adjoint of this functor (when restricted to groups) is the algebra construction above.
  • The concept of a “group with negation” can probably be generalized to that of a “group with \(n\)th roots of unity.” These would have \(\xi^n = 1\) instead of \(\xi^2 = 1\), and we could construct an algebra over \(\mathbb{C}\) by taking the quotient of the group algebra by the ideal generated by \([\xi] - \exp(i\tau/n)[1]\). Such “groups with \(n\)th roots of unity” could probably be characterized as group objects in the category of \(\mathbb{Z}/n\)-sets, with the tensor product being some generalization of the “half-product.”