· Portal  Help Search Members Calendar 
Welcome Guest ( Log In  Register )  Resend Validation Email 
Welcome to Did You Know? 
JaneFairfax 
Posted: 09:00 Tuesday 21 April 2009

The Enlightenment Group: Moderators Posts: 670 Member No.: 20 Joined: 03 Mar 2007 
I have just read the Chapter 14 of A Course in Number Theory by John F. Humphreys, which is about the classification of finite Abelian groups. Man, I had a hard time struggling to follow the difficult proofs of the results presented in this chapter. But the nice thing is that we have a nice result about finite Abelian groups. Did you know? If G is an Abelian group of order n, there exists a unique decomposition of the integer n into its factors as n = The uniqueness is such that if also n = 
algebraic topology 
Posted: 14:45 Tuesday 21 April 2009

Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 
Did you know? If G is a finite Abelian group, then G has a subgroup of order d for every positive integer d dividing the order of G. 
JaneFairfax 
Posted: 17:29 Tuesday 21 April 2009

The Enlightenment Group: Moderators Posts: 670 Member No.: 20 Joined: 03 Mar 2007 
Did you know? The multiplicative group of nonzero elements of a field F is cyclic. For this group is isomorphic to some direct product of cyclic groups for which so that for all nonzero x in F. So x is a root of the polynomial of degree n_{1} in the uniquefactorization domain and so there cannot be more than n_{1} such roots. On the other hand clearly has at least n_{1} elements. Hence and is cyclic. 
algebraic topology 
Posted: 16:29 Wednesday 20 May 2009

Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 
Did you know? The automorphism group of a cyclic group of prime order p is cyclic of order 
JaneFairfax 
Posted: 21:27 Wednesday 20 May 2009

The Enlightenment Group: Moderators Posts: 670 Member No.: 20 Joined: 03 Mar 2007 
Did you know? The automorphism group of the direct product of n copies of the cyclic group of order p is isomorphic to the general linear group of degree n over the field with p elements. 
algebraic topology 
Posted: 01:00 Monday 25 May 2009

Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 
Did you know? The general linear group of degree n over a field with q elements (where q = p^{k} for some prime p and positive integer k) has order 
JaneFairfax 
Posted: 11:33 Monday 25 May 2009

The Enlightenment Group: Moderators Posts: 670 Member No.: 20 Joined: 03 Mar 2007 
Did you know? The special linear group of degree n over the field with q elements is the subgroup of the general linear group consisting of all matrices with determinant 1. This is a normal subgroup and This is because the mapping mapping each matrix to its determinant is an epimorphism of the general linear group onto the multiplicative group of nonzero field elements with the special linear group as its kernel. 
algebraic topology 
Posted: 13:07 Monday 25 May 2009

Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 
Did you know? Hence . 
JaneFairfax 
Posted: 14:19 Monday 25 May 2009

The Enlightenment Group: Moderators Posts: 670 Member No.: 20 Joined: 03 Mar 2007 
Did you know? The projective special linear group of degree n over is the group where is the group of all 
algebraic topology 
Posted: 18:58 Monday 25 May 2009

Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 
Did you know? is simple when 
JaneFairfax 
Posted: 23:57 Monday 25 May 2009

The Enlightenment Group: Moderators Posts: 670 Member No.: 20 Joined: 03 Mar 2007 
Did you know? The Mathieu groups M_{11}, M_{12}, M_{22}, M_{23} and M_{24} are simple. One way of representing these groups is as groups of permutations (which is what Humphreys does) – they are 5 of the 26 socalled sporadic finite simple groups. 
algebraic topology 
Posted: 23:17 Monday 01 June 2009

Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 
Did you know? The Mathieu groups M_{11}, M_{23} and M_{24} are the automorphism groups of the ternary Golay code, the binary Golay code and the extended binary Golay code respectively. 