Page 88 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.

3.2 Basic notions and examples

gap> G := DihedralGroup( 32 );

gap> cs := CompositionSeries( G );
[ Group([ f1, f2, f3, f4, f5 ]), Group([ f2, f3, f4, f5 ]),

Group([ f3, f4, f5 ]), Group([ f4, f5 ]), Group([ f5 ]), Group([ ]) ]
gap> List( [1..5], i -> StructureDescription( cs[ i ] / cs[ i + 1 ] ) );
[ "C2", "C2", "C2", "C2", "C2" ]

The result is not surprising as D32 is a 2-group.

Solvable groups

A finite group is said to be solvable if all of its composition factors are cyclic of prime
order. One can prove the following:

Theorem 3.2.33 A finite group G is solvable if it has a series

G = G0 G1 G2 · · · Gr = {1}

with all Gi /Gi +1 abelian.

The statement of Theorem 3.2.33 is usually taken as the definition of solvable groups
in the infinite case. Every abelian group is solvable. The smallest non-abelian solvable
group is 1 A3 S3. The smallest non-solvable group is A5. The derived length of a solv-
able group G is the length of the shortest abelian series of G . A group is called metabelian
if its derived length is no more than two.

Lemma 3.2.34 The following hold:

1. A subgroup of a solvable group is solvable.
2. A homomorphic image of a solvable group is solvable.
3. If a normal subgroup and its factor are solvable, then the group is solvable.

Lemma 3.2.35 A product of two normal solvable subgroups of a group is again solvable.

PROOF. Let H G and K G be solvable. Then (K H )/K H /(H ∩ K ) is solvable by (2)
above and consequently K H is solvable by (3).

The following shows that A5 is the only non-solvable group of order 60:
gap> l60 := AllSmallGroups( 60 );;
gap> List( l60, IsSolvable );
[ true, true, true, true, false, true, true, true, true, true, true,

true, true ]
gap> notsolv := Filtered( l60, G -> not IsSolvable( G ) );
[ Group([ (1,2,3,4,5), (1,2,3) ]) ]
gap> StructureDescription( notsolv[ 1 ] );
"A5"