Finite Abelian groups, DYK Fact #564

[JaneFairfax]

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…n_r such that n_i > 1 for each i = 1. …, r, n_i+1 divides n_i for each i = 1. …, r−1, and G is isomorphic to (where C_k denotes the cyclic group of order k).

The uniqueness is such that if also n = m₁…m_s such that m_i > 1 for each i = 1. …, s, m_i+1 divides m_i for each i = 1. …, r−1 and G is isomorphic to then s = r and m_i = n_i for all i = 1. …, s.

[algebraic topology]

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]

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 n₁ in the unique-factorization domain and so there cannot be more than n₁ such roots. On the other hand clearly has at least n₁ elements. Hence and is cyclic.

[algebraic topology]

Did you know?

The automorphism group of a cyclic group of prime order p is cyclic of order p−1.

[JaneFairfax]

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]

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 (qⁿ−1)(qⁿ−q)(qⁿ−q²)…(qⁿ−qⁿ⁻¹).

[JaneFairfax]

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]

Did you know?

Hence .

[JaneFairfax]

Did you know? The projective special linear group of degree n over is the group where is the group of all n×n scalar matrices over with determinant 1.

[algebraic topology]

Did you know?

is simple when n = 2 or when n > 1 and q > 3.

[JaneFairfax]

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]

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.