group-theory

Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts Given a pair of groups Grp, or more generally an -tuple of groups, what is traditionally called their free product and denoted or generally is really their coproduct in Grp: (Compare the dual notion of “direct product of groups”, which are really cat…

Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts A finite group is a group whose underlying set is finite. This is equivalently a group object in FinSet. Let be a finite group with order . (Cauchy) If a prime number divides , then equivalently has an element of order ; has a subgroup of order of a …

Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts Given a pair of groups, and , and a joint subgroup in each of their centers, then the corresponding (“external”) central product is the quotient group of the direct product group by the diagonal subgroup . (structural over material definition) Beware…

Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts The permutations of a set form a group, , under composition. This is especially clear if one thinks of the permutation as a bijection on , where the multiplication / composition is just composition of functions. This group is called the symmetric gro…

For permutation groups (or transformation monoids), we don’t need to assume the associative law, since composition of functions is always associative. So a permtation group is a set of mappings satisfying the identity, inverse and closure axioms. This implies that … Continue reading →

When I was a baby undergrad, I was endlessly fascinated by the word group theory. It sounded so innocent, but whenever I opened a book about it, I was confronted with some very scary looking math. But group theory is not all that scary! So in this post, I’d like to explain what a group is (with some pictures). This is the explanation I wish I’d heard when I was first learning it. group theory is …