Jim Caprioli at The Finland Station

Yo! I got an answer to my question about CharacterTables. From silverfish. I find the answer reassuring in the sense that it was written in a very professional style, i.e. the stuff I bumped into just is complicated. That also explains why I couldn't find an answer in the notes, books and websites about the subject. I don't understand the answer ( yet ). It just takes time to study such a (complicated) answer. That's mathematics too. I have something to work on again! The joy of this 'activity' is in that 'something', can't explain it.

Since I am in my blog anyway I found another interesting property worth posting. The characters I did find in Z4x|Z4 were all of dimension 1, these are called linear characters. The number of linear characters is equal to | G / G' | ( in words: "The order of, the quotient group of, the group under investigation and it's derived subgroup"). Think about this for a moment. An abelian group only has linear characters, |G| characters to be precise. Since G/G' is the Abelanization of G it is particularly beautiful that a non-abelian G has |G/G'| linear characters. Some examples. The inevitable S3, and Z4x|Z4 (what else?):

gap> S3:=SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> S3d:=DerivedSubgroup(S3);
Group([ (1,3,2) ])
gap> Size(S3)/Size(S3d);
2
gap> Display(CharacterTable(S3));
CT1

     2  1  1  .
     3  1  .  1

       1a 2a 3a
    2P 1a 1a 3a
    3P 1a 2a 1a

X.1     1 -1  1
X.2     2  . -1
X.3     1  1  1

gap> G:=AllGroups(16)[4];
<pc group of size 16 with 4 generators>
gap> StructureDescription(G);
"C4 : C4"
gap> H:=DerivedSubgroup(G);
Group([ f3 ])
gap> Size(G)/Size(H);
8