Do groups have Duals?

Question

Do groups have Duals?

Might be a bit of a simple question but it should not take too much effort to handle. Notice that I'm not saying all or automatically or anything like that.

I'm merely wondering whether on the level of group and group theory Duals are something you discuss.

Answer 1

Classically, for a group $G$, the dual group of $G$ is defined as $Hom(G, \mu)$, the group of homomorphisms from $G$ to $\mu$ where $\mu$ is the multiplicative group of roots of unity in $\mathbb{C}$.

Answer 2

Let the set $G$ be a group under the operation $\ast.$ The opposite group is the set $G$ with the opposite operation $\circ$ defined by $a\circ b = b\ast a.$ Note that abelian groups are their own opposites.

Obviously, the opposite of the opposite is the original group, so this is a kind of dual operation.

For more, see Mac Lane and Birkhoff, Algebra.

Answer 3

There are a few things claiming to be the dual of a group!

The Pontryagin dual of a locally compact abelian group $G$ is the group of continuous group homomorphisms from $G$ to the multiplicative group of complex numbers with norm $1$, that is, the unit circle in the complex plane. This satisfies the desired property that the dual of the dual is canonically isomorphic to the original group, just as in the vector space case. The Pontryagin dual is used extensively in harmonic analysis, and elsewhere.

The Langlands dual is a substantially more complicated object; the groups it applies to are the reductive algebraic groups, and the definition of "reductive algebraic" is a bit technical. In the Langlands program, the functoriality principle states roughly that a homomorphism between the Langlands duals of two groups should give you a way to transfer a "nice" representation of one to a "nice" representation of the other (I'm not going to try to define this more precisely, since I don't understand it more precisely). Revealing my ignorance, I can't tell you what this is good for - the Langlands program is quite technical, and I don't have a good understanding of what kinds of consequences it has or why the Langlands dual construction is the right construction to get at those consequences. But hopefully someone more knowledgeable than me can chime in.

I believe there are other notions of dual, but these are the two that I've heard of (and the Pontryagin dual is the one I understand).