Page 74 - 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> Order( G );
6
gap> IsCyclic( G );
true
gap> IsAbelian( G );
true

Another way is to represent a cyclic group of order n with a generator x and relation
x n = 1. We first construct a free group on {x } and then factor out the relation x n = 1. For
n = 6, this goes as follows:

gap> F := FreeGroup( "x" );

gap> AssignGeneratorVariables( F );
#I Assigned the global variables [ x ]
gap> G := F / [ x^6 ];

gap> Order( G );
6
gap> StructureDescription( G );
"C6"
gap> Elements( G );
[ , x^3, x^2, x^-1, x^-2, x ]

Note that the groups in the first and second example both represent C6, yet, in GAP’s
eyes they are not identical objects, because GAP represents them in different ways. The
first example represents C6 as a pc group, and the second one as an fp group.

Abelian groups

Finitely generated abelian groups are classified by the following result:

Theorem 3.2.7 (Fundamental Theorem of Abelian Groups) Every finitely generated abe-
lian group is a direct product of cyclic groups

Cm1 × Cm2 × · · · × Cmr × C∞k ,
where mi |mi +1 for all i = 1, . . . , r − 1. Two groups of this form are isomorphic if and only
if the numbers m1, . . . , mr and k are the same for the two groups.

Alternatively, all finite abelian groups are direct products of cyclic groups of prime
power order. This follows from the fact that if m and n are relatively prime then Cm ×
Cn ∼= Cmn . A group that is isomorphic to the direct product of a number of copies of
Cp is called an elementary abelian p -group. Every elementary abelian p -group (written

additively) is also a vector space over GF(p ). The scalar multiplication is given by

λx = x + · · · + x .

λ times