1. ## Groups complex

Hello

I got 2 questions, which seems hard to follow any ideas in how to start off?

Set
S of generators of group G is called a minimal set of generators if
for all
s 2 S, the set S n fsg is no longer a set of generators. Find a minimal

set of generators for
D6

and this one is complex

Use complex numbers to prove that cos
4(beta)+ sin4 (beta)=(1/4)(3 + cos 4(beta)).

Any help is appreciated
Thank you

Belzelga

2. Originally Posted by Belzelga
Hello

I got 2 questions, which seems hard to follow any ideas in how to start off?

Set
S of generators of group G is called a minimal set of generators if
for all
s 2 S, the set S n fsg is no longer a set of generators. Find a minimal

set of generators for
D6

and this one is complex

Use complex numbers to prove that cos
4(beta)+ sin4 (beta)=(1/4)(3 + cos 4(beta)).

Any help is appreciated
Thank you

Belzelga
For the first problem, there are two competing meanings for the notation D_6. It is a dihedral group, with eith 6 elements or 12 elements, according to which definition you use. In either case, it can be described by two generators (a rotation and a reflection). This is a minimal set, because of you remove one generator then the remaining single generator could only generate an abelian group. But whichever way you define the group D_6, it is nonabelian.

I think that the second problem should read $\cos^4\beta + \sin^4\beta = {\textstyle\frac14}(3+\cos(4\beta))$. The way to prove this is to put z=cos(β)+i.sin(β). Then, by de Moivre's theorem, $z^4 = \cos(4\beta) + i\sin(4\beta)$. But $z^4 = (\cos\beta+i\sin\beta)^4 = \cos^4\beta + 4i\cos^3\beta\sin\beta -6\cos^2\beta\sin^2\beta -4i\cos\beta\sin^3\beta + \sin^4\beta$.

Take the real part of those expressions to see that $\cos(4\beta) = \cos^4\beta -6\cos^2\beta\sin^2\beta + \sin^4\beta$. But $\cos^2\beta\sin^2\beta = {\textstyle\frac14}\sin^2(2\beta) = {\textstyle\frac14}\bigl({\textstyle\frac12}(1-\cos(4\beta))\bigr)$. Substitute that into the previous equation and you get the result.