[GAP Forum] Nongenerating elements of a group

[Stefan Kohl]

Dear Forum,

Sandeep Murthy asked:

> I have two questions.
>
> 1. What are the differences between the functions
> GeneratorsSmallest( group ) and MinimalGeneratingSet( group ) ?
> They seem to do the same thing.

The function `GeneratorsSmallest' returns the lexicographically
smallest set of generators of a group, while the function
`MinimalGeneratingSet' returns a generating set of minimal possible
length. Thus, the set of generators returned by the former function
may have not the smallest possible number of elements, while the
set of generators returned by the latter function may be not the
lexicographically smallest set of generators.

-- See the documentation, i.e. ?GeneratorsSmallest and
?MinimalGeneratingSet.

For example we have:

gap> GeneratorsSmallest(SymmetricGroup(4));
[ (3,4), (2,3), (1,2) ]
gap> MinimalGeneratingSet(SymmetricGroup(4));
[ (2,4,3), (1,4,2,3) ]

> 2. Is there a function for testing whether a given element of a
> group is a nongenerating element?

You want to test membership in the Frattini subgroup?
-- This can be done as follows: if G is your group and g is
your given element, check `g in FrattiniSubgroup(G);'.

Hope this helps,

    Stefan Kohl

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Dr. Stefan Kohl

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