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Abstract Algebra (Dummit and Foote) Definitions and Theorems Chapter 5
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Chapter 5 Proposition 1
If G1,...,Gn are groups, their direct product is a group of order |G1| |G2| ... |Gn| (if any Gi is infinite, so if the direct product)
Chapter 5 Proposition 2
Look in the book. Page 154. Too much to write.
5.1 Examples
Look in the book! Page 155.
Definition of a Finitely Generated Group
(1) A group G is finitely generated if there is a finite subset A of G such that G = <A>.
(2) For each r∈ℤ with r>=0, let ℤ^r = ℤ x ℤ x ... x ℤ be the direct product of r copies of the group ℤ, where ℤ^0 = 1. The group ℤ^r is called the free abelian group of rank r.
The Fundamental Theorem of Finitely Generated Abelian Groups
Let G be a finitely generated abelian group. Then:
(1) G is isomorphic to ℤ^r x ℤ_n1 x ℤ_n2 x ... x ℤ_ns, for some integers r, n1, n2, ..., ns satisfying the following conditions:
i) -- r is greater than or equal to 0 and n_j is greater than equal to 2 for all j
ii) -- n_(i+1) divides n_i for 1 <= i <= s-1
(2) The expression in (1) is unique: if G is isomorphic to ℤ^t x Z_m1 x Z_m2 x ... x Z_mu, where t and m1,m2,...,mu satisfy (a) and (b) (i.e., t is greater than or equal to 0 and m_j is greater than equal to 2 for all j and m_(i+1) divides m_i for 1 <= i <= (u-1), then t = r, u = s, and m_i = n_i for all i.
Definition of Free Rank, Invariant Factors of G
The integer r in Theorem 3 (p. 158) is called the free rank of Betti number of G and the integers n1,n2,...,ns are called the invariant factors of G. The description of G in Theorem 3(1) is called the invariant factor decomposition of G.
Chapter 5 Corollary 4
If n is the product of distinct primes, then up to isomorphism the only abelian group of order n is the cyclic group of order n, Z_n.
5.2 Example
Look at Page 160
Chapter 5 Theorem 5
Let G be an abelian group of order n > 1 and let the unique factorization of into distinct prime powers be n = p1^α1 p2^α2 .... pk^αk, then:
(1) G is isomorphic to A1 x A2 x ... x Ak, where |A_i| = pi^αi
(2) Look up the rest on p.161, it will be easier to read.
Definition of Elementary Divisors
The integers p^βj described in Theorem 5.5 (p.161) are called the elementary divisors of G. The description of G in Theorem 5(1) and 5(2) is called the elementary divisor decomposition of G.
Chapter 5 Proposition 6
Let m,n∈ℤ+.
(1) Z_m x Z_n is isomorphic to Z_mn if and only if (m,n) = 1
(2) If n = p1^α1 p2^α2 ... pk^αk then Zn is isomorphic to Z_(p1^α1) x Z_(p2^α2) x ... x Z_(pk^αk).
Definition of Commutator of x and y
Let G be a group, let x, y∈G and let A, B be nonempty subsets of G.
(1) Define [x,y] = x^-1 y^-1xy, called the commutator of x and y.
(2) Define [A,B] = <[a,b] | a∈A, b∈B>, the group generated by commutators of elements from A and from B.
(3) Define G' = <[x,y] | x,y∈G>, the subgroup of G generated by commutators of elements from G, called the commutator subgroup of G.
Chapter 5 Proposition 7
Let G be a group, let x,y ∈ G and let H be a subgroup of G. Then:
(1) xy = yx[x,y] (in particular, xy = yx if and only if [x,y] = 1)
(2) H is normal to G if and only if [H,G] is a subgroup of H.
(3) σ[x,y] = [σ(x),σ(y)] for any automorphism σ of G, G' char G and G/G' is abelian.
(4) G/G' is the largest abelian quotient of G in the sense that if H is normal to G and G/H is abelian, then G' is a subgroup of H. Conversely, if G' is a subgroup of H, then H is normal to G and G/H is abelian.
(5) If φ: G -> A is any homomorphism of G into an abelian group A, then φ factors through G' i.e. G' is a subgroup of kerφ and the following diagram commutes (see it on page 169)
5.4 Examples
See page 171
Chapter 5 Proposition 8
Let H and K be the subgroups of the group G. The number of distinct ways of writing each element of the set HK in the form hk, for some h∈H and k∈K is |H∩K|. In particular, if H∩K = 1, then each element of HK can be written uniquely as a product hk, for some h∈H and k∈K.
Chapter 5 Theorem 9
Suppose G is an abelian group with subgroups H and K such that:
(1) H and K are normal in G
(2) H∩K = 1
Then HK is isomorphic to H x K.
Definition of Internal Direct Product
If G is a group and H and K are normal subgroups of G with H∩K = 1, we call HK the internal direct product of H and K. We shall call H x K the external direct product of H and K (when emphasis is called for).
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