Last updates: 9 February 2010
Let be a subgroup of a group We can try to make the set of cosets of into a group by defining a multiplication operation on the cosets. The only problem is that this doesn't work for the cosets of just any subgroup, the subgroup has to have special properties.
A subgroup of is normal if for each for all
HW:Show that a subgroup of a group is normal iff for all
Let be a subgroup of a group is a normal subgroup of iff with the operation given by is a group.
The quotient group is the set of cosets of normal subgroup of a group with the operation given by
Wow! We actually made this weird set of cosets into a group.
HW: Let be a subgroup of a group Show that is a normal subgroup of
Group homomorphisms are used to compute groups. Let and be groups with identities and respectively.
A group homomorphism is a map between groups and such that
A group isomorphism is a bijective group homomorphism.
Two groups and are isomorphic, if there exists a group isomorphism between them.
Two groups are isomorphic if both the elements of the groups and their operations match up exactly. Think of two groups that are isomorphic as being "the same". When we are classifying groups, we put two two groups in the same class only if they are isomorphic. This is what we mean by classifying groups "up to isomorphism".
HW: Show that if then
Let be a group homomorphism. Let and be the identities for and respectively. Then
The kernel of a group homomorphism is the set where is the identity in .
The image of a group homomorphism is the set
Let be a group homomorphism. Then
Let be a group homomorphism. Let be the identity in Then
Notice that the proof of 2.3 (b) does not use the fact that is a homomorphism, only that it is a function.
HW: Show that if and are any two sets and is a map then iff is surjective.
Suppose and are groups. The idea is to make into a group.
The direct product of two groups and is the set with the operation given by for all We say that the multiplication in is componentwise.
More generally, given groups the direct product is the set given by with the operation given by where and is given by the operation in the group
HW: Show that these are good definitions, ie that as defined above, and are groups with identities given by and respectively ( denotes the identity in the group ).
A group is abelian if for all
The center of a group is the set
HW: Give an example of a non-abelian group.
HW: Prove that every subgroup of an abelian group is normal.
HW: Prove that is a normal subgroup of
HW: Prove that iff is abelian.
The order of a group is the number of elements in
Let be a group and let The order of is the smallest positive integer such that If no such integer exists then
Let be a group and let be a subset of The subgroup generated by is the subgroup of such that
Think of as gotten by adding to exactly those elements of that are needed to make a group.
HW: Let be a group and let be a subsert of Show that the subgroup generated by exists and is unique.
[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)