Examples 1

• $1 \lhd K \lhd A_4 \lhd S_4$ is not a normal series because $K$ is not normal in $S_4$?? $K/1 = C_2$? $V/K = C_2$, $A_4/V = C_3$, $S_4/A_4 = C_2$
• The group $S_5$ is not solvable [washington link]
• The commutator subgroup of the alternating group A₄ is the Klein four group.
• The commutator subgroup of the symmetric group S_n is the alternating group A_n.
• The commutator subgroup of the quaternion group Q = {1, −1, i, −i, j, −j, k, −k} is [Q,Q]={1, −1}.
• A=<a>,B=<b> subgroups of D6 but AB is not a group, because ab is in it but the inverse (ab)^-1 = abab is not.
• Theorem $t = (1\,2)$ and $c = (1\,2\,3\,\ldots\,n)$ generate $S_n$.
  proof: $t^{c^i} = (i\,i+1)$ so we have every transposition. $t^{c^{n-1}}c = (n-1\,n)c = (1\,2\,3\,\ldots\,n-1)$ so we are done by induction.

• http://en.wikipedia.org/wiki/Commutator_subgroup 
• http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/subgpseries1.pdf
• www.math.washington.edu/~greenber/SolvableGroups.pdf