Watched lecture 2 of Abstract Algebra E222

Topics

Example of group: GL(n,R), invertible nXn matrices with elements a_ij taken from R.

Definition of a group
- operation is closed
- operation is associative
- group has identity element
- elements have inverses

Definition of S(n), group of all bijective maps f: S->S, the symmetry group of n elements, with as operation the composition of maps

Definition of a subgroup
- closed
- group has identity element
- elements have inverses

Examples
- S1
- S2
- S3. This example was very messy. Instead of correctly naming the elements e, s, s2, t, st, s2t he named them e, t, t', s, s' and s'', not clearly emphasizing that s' was in fact st and so on. Here he could have nicely drawn a Cayley Table but he didn't.

Definition of transposition as the exchange of only two elements. ( Very important concept )

Example of all the subgroups of Z.
- bZ, all multiples of an integer including {0}, so {+/-2, +/-4, ...}, {+/-3, +/-6,... } are all subgroups.

Proof that all subgroups of Z are of form bZ.
- bZ is a subgroup
- any subgroup is of type bZ
This last part was really excellent, he used the Euclidian division algorithm to complete this part of the proof.

Definition of cyclic subgroup. The smallest subgroup containing an element g. This is the collection {g,g^2,g^3,...}. This subgroup can be either finite or infinite.

Definition of the order of an element g of a group. The smallest positive integer e such that g^m=e.

Next time his colleague / assistent will be lecturing and he will start with Lagranges theorem about that the order of a subgroup is a divisor of the order of a group. So I'll expect cosets will be introduced as well.


Note:

Two surprising, remarkable comments from Benedict Gross of which I am not sure I agree:
- " You cannot learn too much Linear Algebra "
( I agree Linear Algebra is important and fun but imho it will never be able to grasp deep theorems in say Number Theory. I am aware of the importance of Linear Algebra in Group Theory especially Representation Theory )

- " I do not recommend writing out multiplication tables "
( Playing with Cayley Tables gave me definitely more insight in the structure of many groups, if you have Mathematica or GAP producing a Cayley table is not difficult. I doubt if Gross has actual experience with either of the two, or he hides it carefully until later. )