Question

let G = D2n = {e,r,r^2,...,r^n-1,s,sr,sr²,..,sr^n-1} a diedergroup of order 2n, where n >=3
(a) prove that [G,G] = <r²>
(b) prove that G/[G,G] consists of two elements if n is uneven and 4 elements if n is even

Answer 1

The proof for (a) is provided as follows: First we show that every element of the cyclic group <r²> is a commutator.....(1) and later we show that every commutator of G is in the group <r²>...(2), so that (1) and (2) completes the proof.

So let us start with the proof of the fact (1)..The commutator [r,s] is defined as rsr^-1s^-1 which equals rrss^-1. [REASON- In any group (ab)^-1=b^-1a^-1]. That implies [r,s]=r². So r² is indeed a commutator. More generally the above logic can be extended as [r^m,s]=r^msr^-ms^-1=r^mr^mss^-1=r^2m. Thus elements of the type r^2m are commutators which establishes fact 1.

For the other way argument (to establish fact (2) we shall consider elements of the type [a,b] where 'a' and 'b' are either rotations or reflections. [REASON-Dihedral groups are made of rotations and reflections]

If both a and b are rotations then it is a trivial fact that the commutator[a,b] is in <r2>.

If a is a rotation and b is a reflection then a=r^m, b=r^ls for some integral powers m and l. Using the fact that b^-1=b we compute the power aba^-1b^-1=aba^-1b=r^mr^lsr^-mr^ls=r^2m ....[By proper rearrangement]

Similarly if a is a reflection, b is a rotation

(aba^-1b^-1)^-1= bab^-1a^-1. and by the previously made observation this element is also of the type r^2m. and so is (aba^-1b^-1).

Finally if we set both a and b as reflections we get again (aba^-1b^-1) simplified of the type r^2k.

So all the above cases show that every commutator is an element of <r²>.

PROBLEM (b): We are trying to understand the quotient G/[G,G]. If n is uneven, i.e 'n' is odd then by part (a) we see that [G,G] consists of every power 'r' and hence the quotient consists exactly of 2 elements ...It is in fact always abelian. If n is even then the quotient G/[G,G] will be the klein 4 group and hence consists of 4 4 elements.