Give an example of a nonabelian group such that G/Z(G) is... [closed]

Question

A) abelian; B) nonabelian;

Not sure here.

Answer 1

Hint Consider the question for the smallest non-abelian groups, which are the symmetric group $S_3$, the dihedral group $D_4$ and the quaternion group.

Answer 2

A) Consider a dihedral group of symmetries of a triangle. The elements are rotation about its center by $2\pi /4$ radians : $o, o^2, o^3$, reflection about a line through it's center and one of its vertices: $l$ with $l^2 = 1$. Then $ol \neq lo$, check that on paper. So it's not abelian.

According to Wikipedia article, $lol = o^{-1} = o^2$, together with $|\langle o \rangle|= 3$ and $l^2 =1$ is a presentation for the group. So then let's calculate it's order:

It's elements are all strings over $o,l$, st $l$ appears only once consecutively and the preceding presentation rules are adhered to. Then strings of length $1$ are:

$o, l$

Strings of length $2$ are:

$o^2, ol, lo = o^2l$

Strings of length $3$ are: $olo$

(checking each binary number in $o, l$)

Strings of length $4$ are: (using irreducible strings of length 3) There are none.

So counting up we have $|D_3|$ = 6.

Well any quotient groups not isomorphic to $D_3$ are clearly abelian then, since $6$ is sufficiently small. Prove that.