Page 86 - 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

PROOF. By Cayley’s theorem, every group of order n can be embedded as a subgroup of
Sn , and can be generated by k = log2 n elements. There are at most n ! choices for each
g i , so the number of subgroups of Sn is at most

(n !)k ≤ (n n )log2 n = n n log2 n ,

as required.

GAP offers a Small Groups library which gives access to all groups of certain “small”

orders. The groups are sorted by their orders and they are listed up to isomorphism;
that is, for each of the available orders a complete and irredundant list of isomorphism
type representatives of groups is given. The library also has an identification function: it

returns the library number of a given group. More on this can be found in GAP’s manual.

Here are some examples.

gap> AllSmallGroups( 16 );;
gap> NrSmallGroups( 512 );
10494213
gap> AllSmallGroups(Size, 16, IsAbelian, true);
[ ,

,
,
,
]
gap> List( last, StructureDescription );
[ "C16", "C4 x C4", "C8 x C2", "C4 x C2 x C2", "C2 x C2 x C2 x C2" ]
gap> G := DihedralGroup( 64 );

gap> IdGroup( G );
[ 64, 52 ]
gap> H := SmallGroup( 64, 52 );

gap> G = H;
false
gap> StructureDescription( H );
"D64"

3.2.6 Jordan-Hölder theorem

A group G is simple if {1} and G are the only normal subgroups of G . The abelian simple
groups are precisely Cp where p is a prime (exercise). More examples of finite simple
groups will be exhibited in Section 3.3.

A composition series of a group G is a sequence of subgroups

{1} = G0 G1 G2 · · · Gr = G

such that all the factors Gi +1/Gi are simple groups. A related concept is that of chief
series, where Gi are all normal in G and each Gi +1/Gi is a minimal normal subgroup of
G /Gi .