Abstract Algebra [draft] by Justin R. Smith PDF

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If the set X is finite, its order is called the rank of A. This generating set is called a free basis of A. R EMARK . 1 implies that the basis do not satisfy any equations other than those dictated by simple logic (and the fact that the groups are abelian). 11 on page 21) and direct product. 2. Given an infinite set of abelian groups { Ai } (1) their direct product ∞ ∏ Ai i =1 is the set of all infinite sequences ( a1 , a2 , . . ) with ai ∈ Ai . (2) their direct sum ∞ Ai i =1 is the set of infinite sequences ( a1 , a2 , .

1. If S is a set and G is a group, an action of G on S is a map f:G × S → S written as f ( g, s) = g · s with the property that g1 · ( g2 · s ) = ( g1 g2 ) · s for all s ∈ S and all g1 , g2 ∈ G. A group-action will be called transitive if, given any s1 , s2 ∈ S, there exists some g ∈ G such that g · s1 = s2 . R EMARK . The simplest example of this is G acting on itself by multiplication. A more subtle example of a group action is G acting on itself via conjugation: for x, g ∈ G. 2. The symmetric group Sn acts on the set {1, .

9. Shiow that D4 = R, d1 . 10. If G = Z× p for a prime number p, define G2 = x2 x ∈ G Show that G2 ⊂ G is a subgroup of order ( p − 1)/2 if p is odd, and order 1 if p = 2. 11. If g ∈ G is an element of a finite group, show that there exists an integer n > 0 such that gn = 1. 12. Prove that the operation x ∗ y = x + y + xy on the set x ∈ R, x = −1, defines an abelian group. 13. List all of the subgroups of a Klein 4-group. 2. 2. Homomorphisms Now we consider functions from one group to another, and the question of when two groups are mathematically equivalent (even when they are defined in very different ways).

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Abstract Algebra [draft] by Justin R. Smith

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