This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 5125 Friday, September 23 First Test Review The test will cover sections 3.1, 3.2, 3.3, 3.5, 4.14.6, 5.1, 5.2, 5.4, 5.5, 6.3. Topics will include If Z is a central subgroup of G and G / Z is cyclic, then G is abelian. Fundamental homomorphism theorem and related isomorphism theorems. Group actions,  G x  G x  =  G  If G has a subgroup H of index n , then there exists K G such that K H and  G / K  n !. Class equation The center of a nontrivial pgroup is nontrivial. Groups of order p 2 are abelian. Sylow theorems. If G is a simple group which contains a subgroup of index n and  G  6 = 2, then G is isomorphic to a subgroup of A n . Proving groups of certain orders cannot be simple. A n is simple for n 5 Fundamental structure theorem for finitely generated abelian groups; elementary di visors and invariant factors Semidirect products Free groups, universal property of free groups Solution to Problem 1 on Sample Test no. 1 We consider the conjugation action of G on G given by g x = gxg 1 for g , x G . Since the order of x is the same as the order of gxg 1 , elements of order two are sent to elements of order two under this action and therefore the size of an orbit containing an element of order two is 1, 2 or 3. Lets consider separately each of these possibilities (its worth noting at this stage that G must have at least 4 elements). Suppose { x } is an orbit of size one. This means that gxg 1 = x for all g G and thus x is in the center of G . It follows easily that h x i is a nontrivial normal subgroup of G and so G is not simple. Suppose there is an orbit of size two. Then the stabilizer of an element in this orbit will have index two in G . Since subgroups of index two are always normal, G has a nontrivial normal subgroup (because...
View
Full
Document
 Fall '07
 PALinnell
 Math, Algebra

Click to edit the document details