AA VI, VII, VIII & IX

AA VI, VII, VIII & IX

1. First Theorems

   • Let (G,◦) be a group. Then (G,◦) has a unique identity element.
   • Let (G,◦) be a group and let a be an element of G. Then a has a unique inverse.
   • Let G be a group, and let a ∈ G. Then an ∈ G for all n ∈ Z. Moreover, if m, n are integers then (a^m)^n = a^(mn) , a^m*a^n = a^(m+n). Further, if the group G is abelian, a, b ∈ G and n an integer then (ab)^n = a^n*b^n
   • Let G be a group and a, b ∈ G. Then (ab) ^1 = b^−1*a^−1
   • Let G be an (abelian) group with addition as the binary operation, and let a ∈ G. Then na ∈ G for all n ∈ Z. Moreover, if m, n are integers then m(na) = (mn)a, ma +na = (m +n)a. Further, if a, b ∈ G and n an integer then n(a +b) = na +nb.

2. More examples of groups

   • GL2(R) is group under multiplication of matrices. We call GL2(R) the general linear group
   • Let m be an integer satisfying m ≥ 2. Then (Z/mZ,+) is an abelian group

3. Orders and Lagrange's Thm

   • The order of an element a in a group G is the smallest positive integer n such that a n = 1. If there is no such positive integer n, we say a has infinite order
   • Let G be a group and g be an element of G. (i) g has order 1 if and only if g is the identity element. (ii) Let m be a non-zero integer. Then g m = 1 if and only if g has finite order d with d | m.
   • Let G be a group. The order of G is the number of elements that G has. We denote the order of G by |G| or #G
   • (Lagrange’s Theorem—Version 1) Let G be a finite group, and let g be an element of G. The order of g divides the order of G
     • Let G be a finite group of order n, and let g be an element of G. Then g^n = 1

4. Subgroups

   • Let (G,◦) be a group. Let H be a subset of G and suppose that (H,◦) is also a group. Then we say that H is a subgroup of G (or more formally (H,◦) is a subgroup of (G,◦)). For H to be a subgroup of G, we want H to a group with respect to the same binary operation that makes G a group.
     • Let G be a group. A subset H of G is a subgroup if and only if it satisfies the following three conditions (a) 1 ∈ H, (b) if a, b ∈ H then ab ∈ H, (c) if a ∈ H then a^−1 ∈ H.
       • Before you start answering: You might be wondering why I don’t specify the binary operation on G. Mathematicians generally don’t; you’re expected to figure it out from the context. In maths it is good to play the rôle of the devil’s advocate, but not to the extent of renouncing good taste and common sense. Yes, there are binary operations other than addition that make the set of real numbers into a group. But if I wanted anything other than the usual or obvious operation I’d have told you so.
   • Un is a subgroup of C* of order n
     • The notation Un is not standard. Why do I point this out? You must always be careful with notation: do other people understand you? If you write C ∗ then this is standard notation and every mathematician will know what you mean.
   • SL2(R) = {A ∈ M2×2(R) : det(A) = 1}
     • We define SO2(R) = {Rθ : θ ∈ R}. This is called the special orthogonal group. SO2(R) is a subgroup of GL2(R)
       • It is clear (at least geometrically) that RθRφ = RφRθ. Thus SO2(R) is an abelian subgroup of the non-abelian group GL2(R).
   • Let C be the set of infinitely differentiable real functions. It is a fact that C is an additive group
     • An example of a subgroup of this is H = {at 2 : a ∈ R}.
   • Non-Trivial and Proper Subgroups
     • Let G be a group. Then G and {1} are subgroups.
     • (Lagrange’s Theorem—Version 2) Let G be a finite group, and H a subgroup of G. Then the order of H divides the order of G