Thurs March 2
* old homework:
RF rubik group.
- number of positions ?
- operators ?
full rubik group.
- discussion ?
- (Jim's version online : still coming)
* Lagrange Theorem: (Dog school notes, chap 8)
"The order of a subgroup H of a group G divides the order of G."
Proof:
(a) define "coset" : given a subgroup H=(h1,h2,...) and an
element x not in H, (x.h1, x.h2, x.h3, ...) is a "left coset"
of the subgroup H with respect to x.
(b) lemma 1: each coset of H has the same number of elements of H.
(c) lemma 2: any two cosets of H are either dijoint or identical.
(d) then H divides G into a number of disjoint sets, each with
ord(H) elements. Thus ord(G)/ord(H) must be an integer, namely
the number of elements in each coset.
OR
Show that "y1 and y2 are in the same coset" is an equivalence
relation, and this equivalence relation partitions G into
disjoint sets, each of the same size. The result follows.
Corollary 1:
If N is cycle length of x, i.e. x^N=identify, N must divide ord(G).
Corollary 2:
All groups of prime order are cyclic.
* converse??
If ord(G) is a multiple of a number N, must G have a subgroup of order N?
Can you find a counterexample?
* Enumeration of finite groups:
order 1 1 group, C1
2 1 C2
3 1 C3
4 2 groups: C4, D2 - something to do with factors of 4...
5 1 group
6 2 groups C6, S3=D3
7 1 group
8 ?
... ? is there a pattern here?
* Final projects
Assignment:
* How many distinct groups are there of order 8?
Name and/or describe them.
* Finish RF algorithms if you haven't already
* Initial proposal for final project
* Due after break: write up a "solution" to Rubik's Cube,
or at least as much of it as you can manage.
Coming after break: plane figures, wall-paper, Escher