[r-t] Group Theory again

Richard Smith richard at ex-parrot.com
Thu Feb 14 09:31:59 UTC 2013

Robin Woolley 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?

The algorithm is easy enough to explain.  Written formally, 
if H and K are subgroups of G, then <H,K> is the smallest 
subgroup of G containing both H and K as subgroups.

What this means is that if you take a set of generators for 
H (any generator set will do) and a set of generators K, 
then the group <H,K> is the group generated by all the 
generators of H *and* all the generators for K.

> 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.

Define "not obvious".  It's true, anyway.

But it seems fairly obvious to me that <1423,2413> must be 
S_4.  Clearly the order of the group must be a multiple of 
the the l.c.m. of 3 and 4, which is 12.  So either the group 
is A_4 or it is S_4.  As you have an odd-parity row (the 
generator 2413, for example), it cannot be A_4.  Therefore 
it must be S_4.


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