[r-t] Group Theory again

[Martin Bright]

On 14 February 2013 10:53, Robin Woolley <robin at robinw.org.uk> wrote:
> Today's request for help: I get the impression that, given two subgroups, it is  a non-trivial problem to find the smallest subgroup containing both. Is this correct? Example, consider S4 and subgroups generated by <1423> and <2413> of order 3 and 4 respec. It is not obvious, but it seems that the smallest group containing these elements is S4 itself.

It rather depends on what you mean by "finding" a subgroup.  You
probably specify a subgroup by a set of elements which generate it.
So your problem is trivial - given two subgroups specified by
generators, you can "find" the smallest subgroup containing both by
lumping all the generators together.  So now you realise that the real
problem is, given two subgroups given by generators, to decide whether
they're the same, and this comes down to the problem of determining
whether a given element lies in the subgroup generated by a given set
of elements.  That, in general, is a bit tricky, though any decent
computer algebra system can do it very quickly for the size of groups
you're talking about.

Martin