Last updates: 9 February 2010
We start with some basics, just a set and one operation. We can think of the operation as addition or multiplication, or something else, like a composition of functions.
A group is a set and an operation (we write as for ) such that
A subgroup of a group is a subset such that
The trivial group is the set containing only with the operation given by
HW: Show that if is a group, then the identity element of is unique.
HW: Show that if then the inverse is unique.
HW: Why isn't a group?
Given such a definition the next step is to find out what kinds of structures fit the definition and explore them. Examples of groups are
Let be a group and set be a subgroup of . We will use the subgroup to divide up the group .
Unless we specify otherwise we shall always work with left cosets and just call them cosets.
HW: Let be a group and let be a subgroup of . Let and be two elements of Show that iff
Let be a group and let be a subgroup of Then the cosets of in partition
Let be a group and let be a subgroup of Then, for any
Let be a subgroup of a group Then
The above results show that the cosets of a subgroup divide the group into equal size pieces, one of these pieces being the subgroup itself.
A set of coset representatives of in is a set of distinct elements of such that
The index of a subgroup in a group is the number of cosets of in
HW: Show that
[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)