Cohomology and Extensions
A group is a monoid in which every element has an inverse (necessarily unique).
An abelian group is a group in which moreover the order in which two elements are multiplied is irrelevant.
To some extent, a group “is” a groupoid with a single object, or more precisely a pointed groupoid with a single object.
The delooping of a group is a groupoid with
Since for two groups, functors are canonically in bijection with group homomorphisms , this gives rise to the following statement:
Let Grpd be the 1-category whose objects are groupoids and whose morphisms are functors (discarding the natural transformations). Let Grp be the category of groups. Then the delooping functor
is a full and faithful functor. In terms of this functor we may regard groups as the full subcategory of groupoids on groupoids with a single object.
It is in this sense that a group really is a groupoid with a single object.
But notice that it is unnatural to think of Grpd as a 1-category. It is really a 2-category, namely the sub-2-category of Cat on groupoids.
And the category of groups is not equivalent to the full sub-2-category of the 2-category of groupoids on one-object groupoids.
The reason is that two functors:
coming from two group homomorphisms are related by a natural transformation with single component for each element such that the homomorphisms and differ by the inner automorphism
To fix this, look at the category of pointed groupoids with pointed functors? and pointed natural transformations. Between group homomorphisms as above, only identity transformations are pointed, so becomes a full sub--category of (one that happens to be a -category). (Details may be found in the appendix to Lectures on n-Categories and Cohomology and should probably be added to pointed functor? and maybe also k-tuply monoidal n-category.)
A group object internal to a category with finite products is an object together with maps , , and such that various diagrams expressing associativity, unitality, and inverses commute.
Equivalently, it is a functor whose underlying functor is representable.
For example, a group object in Diff is a Lie group. A group object in Top is a topological group. A group object in Sch/S (the category or relative schemes) is an -group scheme. And a group object in , where CAlg is the category of commutative algebras, is a (commutative) Hopf algebra.
A group object in Grp is the same thing as an abelian group (see Eckmann-Hilton argument), and a group object in Cat is the same thing as an internal category in Grp, both being equivalent to the notion of crossed module.
In higher categorical and homotopical contexts
Internalizing the notion of group in higher categorical and homotopical contexts yields various generalized notions. For instance
And the notion of loop space object and delooping makes sense (at least) in any (infinity,1)-category.
Notice that the relation between group objects and deloopable objects becomes more subtle as one generalizes this way. For instance not every group object in an (infinity,1)-category is deloopable. But every group object in an (infinity,1)-topos is.
Following the practice of centipede mathematics, we can remove certain properties from the definition of group and see what we get:
- remove inverses to get monoids, then remove the identity to get semigroups;
- or remove associativity to get loops, then remove the identity to get quasigroups;
- or remove all of the above to get magmas;
- or instead allow (in a certain way) for the binary operation to be partial to get groupoids, then remove inverses to get categories, and then remove identities to get semicategories
Special types and classes
Standard examples of finite groups include
Standard examples of non-finite groups include
Standard examples of Lie groups include
Standard examples of topological groups include
For more see counterexamples in algebra.
A non-abelian group, all of whose subgroups are normal:
A finitely presented, infinite, simple group
Thomson's group T.
A group that is not the fundamental group of any 3-manifold.
Two finite non-isomorphic groups with the same order profile.
A counterexample to the converse of Lagrange's theorem.
The alternating group has order but no subgroup of order .
A finite group in which the product of two commutators is not a commutator.
monoid, monoid object,
group, group object
ring, ring object
automorphism group, automorphism 2-group, automorphism ∞-group,
center, center of an ∞-group,
inner automorphism group
outer automorphism group, outer automorphism ∞-group