When you are not being able to study math

I am in England for a BPEL course. Haven't done much math for the last three days. It makes me feel empty. Math is like playing an instrument. You must practice every day. It is bearable because just before I left I had this idea which I can freely think about anytime, anywhere.

Transforming ideas to results...

A drawing of an idea. How much time would it cost me to convert a group presentation like 'a4+1' to an Excel sheet containing the corresponding Cayley Table? - I certainly would like to have such a program ( coded by myself ). But will I hold on to the idea long enough?

Cayley table of the group ( C3 X C3 ) : C2

I constructed the Cayley table of the non-abelian group ( C3 X C3 ) : C2 which is the semi-direct product of the abelian groups C3 X C3 and C2.

Presentation of the group ( C3 X C3 ) : C2

{a,b,c | ab=ba, ac=c^2a, bc=c^2b} = (C3XC3):C2

Theorem about normal subgroups

Normal subgroups are very important objects in Group Theory. One of the 'must-never-forget'-theorems is the following.

Let N be a normal subgroup of a group G and H be any subgroup of G. Then the intersection of H and N is a normal subgroup of H.

For the proof we use the following theorem.
A subgroup H is a normal subgroup in G if gH=Hg for all elements g in G.



In easy to remember math: "The intersection of a normal subgroup with another subgroup is normal in that subgroup." ( H&N is N(ormal)in H )

If this seems difficult: this theorem becomes trivial real fast.

Morphisms

Let G,H be groups and x,y elements of G.

A homomorphism is a mapping
f: G-> H
such that
f(1) = 1, and
f(xy) = f(x)f(y).

-morphism	when f is
Mono-	injective
Epi-	surjective
Iso-	bijective
Endo-	any and when H=G
Auto-	bijective and when H=G

Math Video

Blogger has been notified, according to the terms of the Digital
Millennium Copyright Act (DMCA), that content in your blog mathematics-diary.blogspot.com allegedly infringes upon the copyrights of others.

( This post contained a link to a torrent of the video Joy of Mathematics. )

Math Speak

There are circles where this is common speak, really:

"How many consecutive digits of pi (3.1415 . . . ) can you display with a deck of cards?"

But it is the sort of question Charlie Eppes might have asked when he was eight years old or so.

Groups of Finite Order By Robert D. Carmichael

I have found a beautiful book on Group Theory which was first published in 1937. I am not at all surprised that it was reprinted in 2000.


Link to Google Books

It is over 440 pages and contains many interesting exercises. I am going to try to solve the following question entirely with the Mathematica Abstract Algebra add-on package.



( But more on this interesting question later. )