Groups

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Groups are one-object categories for which all morphisms are iso. Spelling this out, a groups is a set of maps, typically called \(a, b, c\ldots\), including an identity typically written as \(1\) or \(e\). The maps can be composed to get e.g. \(a\circ b\) (more typically written as \(ab\)), and inverted (\(a^{-1}\)), such that \(aa^{-1}=1\).

Groups are simple enough to be quite well understood, and appear every when we want to think of transformations on a space.

Category of groups

GRP, the category of groups, has groups as objects, and group homomorphisms as maps.

The kernel of a map \(f : A \to B\) is the subset of \(A\) (which turns out to be a sub*group*) such that for \(a \in A, f(a) = 1 \in B\).

Subgroups and Quotients

A subgroup is what you'd expect: as with a subspace of a linear map, recall that closure is important. That is, if \(a,b \in H\), and \(H\) is a subgroup of \(G\), then \(ab\) is in \(H\) too.

A subgroup \(N \subset G\) is normal iff for \(\forall n\in N, \forall g\in G, gng^{-1} \in N\).

A quotient \(G/\sim\) is the space of equivalence classes of a space \(G\) under some equivalence relation \(\sim\).

In particular, you can quotient a group by the equivalence relation implied by a normal subgroup. E.g. for \(N \subset G\), and \(a,b\in G\), let \(a \sim b\) if \(\exists n \in N, an = b\).

First isomorphism theory

For groups \(G\) and \(H\), and \(f : G \to H\):

• the kernel of \(f\) is a normal subgroup of \(G\)
• the image of \(f\) is a subgroup of \(H\)
• the image of \(f\) is isomorphic to the \(G / ker(f)\)

Under construction