# Is there a name for a group where elements either commute or anti-commute?

An abelian group is a group where each element commutes with any other element in the set, so $$ab=ba$$.

I was reading the wikipedia article on the anticommutative property. I think this could be generalised to arbitrary groups as follows:

Suppose $$G$$ embed into the group of units of some ring $$R$$ (so for $$g\in G$$ the element $$-g$$ makes sense, as $$-1$$ is a unit). Then $$a, b\in G$$ anticommute if $$ab=-ba$$.

Another, possibly more general, interpretation is:

Let $$-1$$ be a distinguished central element of order two in $$G$$ (so for $$g\in G$$ the element $$-g=g(-1)=(-1)g$$ makes sense). Say that $$a, b\in G$$ anticommute with respect to $$-1$$ if $$ab=-ba$$.

I was therefore wondering: Is there a specific name for a group where any two elements in the group either commute or anti-commute (with respect to a fixed element $$-1$$)? That is, for $$a,b$$ in $$G$$ either $$ab=ba$$ or $$ab=-ba$$.

• How do you define anti-commute in a group? – J. W. Tanner Aug 31 '20 at 2:40
• What is $-1$ in a group? – Robert Israel Aug 31 '20 at 2:51
• Perhaps you mean $a * b = (b * a)^{-1}$? – Robert Israel Aug 31 '20 at 2:52
• And what is the $-$ in $-(b \ast a)$? The inverse? $(b\ast a)^{-1}$ would be more common notation. – Nate Eldredge Aug 31 '20 at 2:52
• Note that if $(a\star b)=(b\star a)^{-1}$ for all $a,b$ then $g^2=1$ for all $g$, so $g^{-1}=g$ for all $g$, so $(a\star b)=(b\star a)$; an 'anticommutative' group is a commutative group. – Steven Stadnicki Aug 31 '20 at 2:54

It's possible to make sense of this question as follows. Let $$G$$ be a group together with a distinguished central element of order $$2$$ which we call $$-1$$; we'll write the product $$(-1)a$$ as $$-a$$. Say that two elements $$a, b \in G$$ anticommute if $$ab = -ba$$. Then we have, more or less by definition:
Claim: Every pair of elements in $$G$$ either commutes or anticommutes iff the quotient $$G/\{ 1, -1 \}$$ is abelian.
So the desired groups are precisely the central extensions of abelian groups by $$C_2$$. These groups are 2-step nilpotent, and in the finite case every such group must have the form $$G = G_1 \times G_2$$ where $$G_1$$ is an abelian group of odd order and $$G_2$$ is a central extension of an abelian $$2$$-group by $$C_2$$.
The two smallest nonabelian examples of such groups are the quaternion group $$Q_8$$ and the dihedral group $$D_8$$, which are central extensions of $$C_2 \times C_2$$ by $$C_2$$. See the Wikipedia article on extraspecial groups for more. $$D_8$$ is also isomorphic to the Heisenberg group $$H_3(\mathbb{F}_2)$$.
The group algebra $$k[G]$$ of such a group has a distinguished quotient where we identify $$-1 \in G$$ with the actual element $$-1 \in k[G]$$ (apologies for the abuse of notation); this is a certain twisted group algebra of $$G / \{ 1, -1 \}$$. Applying this construction to the quaternion group $$Q_8$$ over $$k = \mathbb{R}$$ produces the quaternion algebra $$\mathbb{H}$$.