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. )

Displaying Cayley Tables in Mathematica

( The structure of the Ring Z[5]: addition and multiplication mod 5. )

0.x = 0 is not an axiom

The statement 0.x = 0 is not an axiom and can thus be proved.

The axioms for the integers are, for addition
(A1) a + (b + c) = (a + b) + c
(A2) a + 0 = a
(A3) a + (-a) = 0
(A4) a + b = b + a
for multiplication
(M1) a(bc) = (ab)c
(M2) 1a = a
(M3) ab = ba
and for addition and multiplication ('distributive laws')
(D1) a(b+c)=ab+ac
(D2) (a+b)c=ac+ab.

So how do mathematicians prove that 0.a=0?
They do something like this.
a = a
a = (1+0).a (by A2 and M2)
a = 1.a + 0.a (by D2)
a = a + 0.a (by M2)
0.a = 0 by (A2).

The field of fractions of an integral domain

Create a set V of ordered pairs from {..., -2, -1, 0, 1, 2, ...} (integers) and {..., -2, -1, 1, 2, ...} (integers excluding 0). Elements of V are for example (3,1), (5,1) and (4,2).

Create an equivalence relation on elements of V. Two elements (a,b) and (c,d) are 'equivalent', 'belong to the same equivalence class' if ad=bc. For example (4,2) and (8,4) are equivalent while (4,1) and (8,4) are not.

Define addition '+' as (a,b) + (c,d) = (ad + bc, bd).

Define multiplication '.' as (a,b) . (c,d) = (ac, bd).

This is how the field of Rationals is formally constructed from the Integers.

Math is more than this...

( How very cliche to put a Mandelbrot fractal in a Math blog. )

The differentiation matrix for arithmetic polynomials

Calculating a difference function is a straightforward process:





But it is simpler to use the differentiation matrix for arithmetic polynomials:



The 5x5 matrix above is suitable for polynomials up to degree 4. It is possible to create a (n+1)x(n+1) matrix capable of handling polynomials up to degree n.

Proof:
Exercise (hint: use falling powers).

Question: Is there a compact way ( recursive, perhaps ) of describing the matrix capable of handling polynomials up to degree n?