7.55 IsCyclic

IsCyclic( G )

IsCyclic returns true if G is cyclic and false otherwise.

A group G is cyclic if and only if there exists an element g in <G> such that G is generated by g.

Note that IsCyclic sets and tests the record component G.isCyclic.

    gap> z6 := Group( (1,2,3), (4,5) );;
    gap> IsCyclic( z6 );
    true
    gap> z36 := AbelianGroup( AgWords, [ 9, 4 ] );;
    gap> IsCyclic( z36 );
    true 

The default function GroupOps.IsCyclic returns false if G is not an abelian group. Otherwise it computes the abelian invariants (see AbelianInvariants) if G is infinite. If G is finite of order p_1^{e_1} ... p_n^{e_n}, where p_i are distinct primes, then G is cyclic if and only if each <G>^{p_i} has index p_i in G.

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