The first mathematician to use the term “group” was Évariste Galois (according to the book A Course in Group Theory by John F. Humphreys). At about that same time, Augustin Louis Cauchy was independently studying these mathematical structures, but it was Galois’s terminology which became universally adopted.

http://z8.invisionfree.com/DYK/index.php?showtopic=118

Last edited by JaneFairfax (2009-04-15 03:12:59)

Last edited by JaneFairfax (2009-04-02 05:52:31)

JaneFairfax wrote:

   

Ricky wrote:

Indeed, this is a rather cool result.  In fact, it generalizes fairly well.  That is, if G contains a subgroup H of index p for the smallest prime dividing the order of G, then H is normal in G.

Do you know how to prove this?

I’ve just read it A Course in Group Theory by John F. Humphreys! It is stated as Corollary 9.25 on page 86.

I didn’t know how to prove it, so I quote Humphreys’s proof.

Apply Corollary 9.23 to the subgroup H to find a normal subgroup N with |G : N| dividing p! Since |G : N| also divides |G|, it divides the greatest common divisor of p! and |G|. Since p is the smallest prime dividing |G|, the greatest common divisor of p! and |G|, so |G : N| = p = |G : H|. Since N is contained in H, it follows that N is equal to H.

states that there is a normal subgroup

with

and

. In fact,

is the kernel of a homomorphism mapping

to the permutation group of the set of all left cosets of

in

.

I have read every proof in my book carefully, and have followed every proof so far. I can thus safely say that I haven’t missed any trick, big or small.

Last edited by JaneFairfax (2009-04-15 03:13:17)