Jim Caprioli at The Finland Station

I am interested in Group Theory since early this year when I discovered the Mathematica package Abstract Algebra. When I looked at that package I was convinced that it would open some doors to rooms of mathematics I hadn't entered before. And it did. From what I knew from mathematics, Number Theory fascinated me most. I try to follow the developments regarding Prime Numbers . It turns out that the classification of all groups is determined by prime numbers. Many theorems in Group Theory are connected to Number Theory.

When I was working on the exercise to describe the character table of a Non-Abelian group of order 21, I was at first surprised there was a Non-Abelian Group of order 21 at all, I expected to find only one group: C21 ( isomorphic to C3 x C7 ). But there was this C3 : C7. The semi-direct product between C3 and C7. I wanted to know why this was the case. I looked at groups with an order divisible by 3 if they were the semidirect product of C3 and some other group:

- 9
- 12 C3 : C4
- 15 -
- 18 -
- 21: C3 : C7
- 24: C3 : C8, C3 : Q8
- 27: C3 : (C3 x C3)
- 30: -
- 33: -
- 36: -
- 39: C3 : C13
- 42: -
- 45: -
- 48: C3 : (C4 x C4)
- 51: -
- 54: -
- 57: C3 : C19
- 60: -
- 63: -
- 66: -
- 69: -
- 72: -
- 75: C3 : (C5 x C5)
- 78: -
- 81: C3 : (C3 x C9), and more...
- 84: C3 : (C4 x C7)
- 87: -
- 90: -
- 93: C3 : C31
- Sofar...

My intuition told me that I should look closer at the charactertables of groups with order 18n+3, n=1,2...
- 21: C3:C7, 1-1-1-3-3 ( first column character table)
- 39: C3:C13, 1-1-1-3-3-3-3
- 57: C3:C19, 1-1-1-3-3-3-3-3-3
- 75: C3:(C5xC5), 1-1-1-3-3-3-3-3-3-3-3
- 93: C3:C31,1-1-1-3-3-3-3-3-3-3-3-3-3
- 111:C3:C37,1-1-1-3-3-3-3-3-3-3-3-3-3-3-3
- 129:C3:C43,1-1-1-3-3-3-3-3-3-3-3-3-3-3-3-3-3
- 147:C3:C49,1-1-1-3-3-3-3-3-3-3-3-3-3-3-3-3-3-3-3
- 165: - :-( ( Pattern breaks here.
- 183:C3:C61,1-1-1-3-3-3-3-3-3-3-3-3-3-3-3-3-3-3-3-3-3-3-3
Notice that ( for example ) 21 = 3 * 1^2 + 2 * 3^2, 57 = 3 * 1^2 + 6 * 3^2, etc. The sum of the squares of the numbers in the first column of the charactertable is equal to the order of the group.

Isn't this beautiful? Magical? Mysterious? Thrilling? Something is out there. Group theory is not an invention of mankind, it is a tool to understand symmetries in nature. All these patterns are determined by prime numbers. Prime numbers are everywhere. This must look childish one day... well, if it does I have gained more insight.