charlamov said:
how many groups of order 24 are there? i need upper estimate with explanation. i know there are 15 of them but i need upper estimate with explanation, explanation is most important. maybe there can be useful that there are 5 groups of order 8.
As there are \,p(3)=3\, partitions of 3, and \,24=2^3\cdot 3\, , there are 3 non-isomorphic groups of order \,24\,.
As there are \,3\, non-isomorphic non-abelian groups of order 8 there are at least 3 nonabelian groups of order 24, each containing one the above
groups of order 8 as Sylow 2-subgroup.
Now, I guess you can pick up some of them on the air: S_3\times C_4\,\,,\,S_3\times C_2\times C_2\,\,,\,A_4\times C_2\,\,,\,Q_8\times C_3\,\,,... .
Check all these are non-isomorphic.
Of course, one can continue with semidirect products...for example, suppose there's a unique Sylow 2-subgroup \,P\, , which is then normal, so
that we can take one of the Sylow 3-sbgsp. \,Q\, and make it act on \,\operatorname{Aut}(P)\, by conjugation: q\cdot x=: x^q=q^{-1}xq\,\,,\,q\in Q\,\,,\,x\in P
It's not specially hard to see that this is a non-trivial action iff the group is non-abelian (which we can assume as the abelian
groups are already sorted out) and this gives us a non-abelian group different from the ones listed above.
In case \,P\, is abelian we can also assume \,Q\, is non normal.
DonAntonio
Ps The above is not, of course, an exhaustive listing neither of all the possible groups of order 24 nor of all the different constructions to ge them.
