Finite Abelian groups, DYK Fact #564
 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 = n1n2…nr such that ni > 1 for each i = 1. …, r, ni+1 divides ni for each i = 1. …, r−1, and G is isomorphic to $C_{n_1}\times\cdots\times{C_{n_r}}$ (where Ck denotes the cyclic group of order k). The uniqueness is such that if also n = m1…ms such that mi > 1 for each i = 1. …, s, mi+1 divides mi for each i = 1. …, r−1 and G is isomorphic to $C_{m_1}\times\cdots\times{C_{m_s}},$ then s = r and mi = ni for all i = 1. …, s.
 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 $C_{n_1}\times\ldots\times C_{n_r}$ for which $n_{i+1}\,\mid\,n_i$ so that $x^{n_1}=1_F$ for all nonzero x in F. So x is a root of the polynomial $x^{n_1}-1_F$ of degree n1 in the unique-factorization domain $F[x]$ and so there cannot be more than n1 such roots. On the other hand $F\setminus\{0_F\}$ clearly has at least n1 elements. Hence $|F\setminus\{0_F\}|=n_1$ and $F\setminus\{0_F\}$ 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 p−1.
 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 = pk for some prime p and positive integer k) has order (qn−1)(qn−q)(qn−q2)…(qn−qn−1).
 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 $\|SL_n(\mathbb{F}_q)\|\:=\:\frac{(q^n-1)(q^n-q)\,\cdots\,(q^n-q^{n-1})}{q-1}$.
 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 $\mathbb{F}_q$ is the group $SL_n(\mathbb{F}_q)\,/\,Z_n(\mathbb{F}_q)$ where $Z_n(\mathbb{F}_q)$ is the group of all n×n scalar matrices over $\mathbb{F}_q$ with determinant 1.
 algebraic topology Posted: 18:58 Monday 25 May 2009 Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 Did you know? $PSL_n(\mathbb{F}_q)$ is simple when n = 2 or when n > 1 and q > 3.
 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 M11, M12, M22, M23 and M24 are simple. One way of representing these groups is as groups of permutations (which is what Humphreys does) – they are 5 of the 26 so-called 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 M11, M23 and M24 are the automorphism groups of the ternary Golay code, the binary Golay code and the extended binary Golay code respectively.
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