By Paul T. Bateman

My target is to supply a few assist in reviewing Chapters 7 and eight of our publication summary Algebra. i've got incorporated summaries of every one of these sections, including a few basic reviews. The assessment difficulties are meant to have rather brief solutions, and to be extra ordinary of examination questions than of ordinary textbook exercises.By assuming that it is a evaluation. i've been capable make a few minor alterations within the order of presentation. the 1st part covers quite a few examples of teams. In offering those examples, i've got brought a few thoughts that aren't studied until eventually later within the textual content. i believe it's priceless to have the examples gathered in a single spot, that you should confer with them as you review.A entire checklist of the definitions and theorems within the textual content are available on the net web site wu. math. niu. edu/^beachy/aaol/ . This website additionally has a few crew multiplication tables that are not within the textual content. I should still be aware minor alterations in notation-I've used 1 to indicate the id component to a gaggle (instead of e). and i have used the abbreviation "iff" for "if and merely if".Abstract Algebra starts on the undergraduate point, yet Chapters 7-9 are written at a degree that we contemplate acceptable for a pupil who has spent the higher a part of a yr studying summary algebra. even though it is extra sharply centred than the traditional graduate point textbooks, and doesn't move into as a lot generality. i am hoping that its good points make it an excellent position to profit approximately teams and Galois thought, or to study the fundamental definitions and theorems.Finally, i need to gratefully recognize the help of Northern Illinois college whereas scripting this assessment. As a part of the popularity as a "Presidential instructing Professor. i used to be given depart in Spring 2000 to paintings on initiatives with regards to educating.

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**Extra resources for Abstract Algebra: Review Problems on Groups and Galois Theory**

**Example text**

We can give the full story for Galois groups of finite fields. We use the notation Aut(F ) for the group of all automorphisms of F , that is, all one-to-one functions from F onto F that preserve addition and multiplication. The smallest subfield containing the identity element 1 is called the prime subfield of F . If F has characteristic zero, then its prime subfield is isomorphic to Q, and if F has characteristic p, for some prime number p, then its prime subfield is isomorphic to Zp . In either case, for any automorphism φ of F we must have φ(x) = x for all elements in the prime subfield of F .

In a group G, any element of the form xyx−1 y −1 , with x, y ∈ G, is called a commutator of G. Beachy 37 (a) Find all commutators in the dihedral group Dn . Using the standard description of Dn via generators and relations, consider the cases x = ai or x = ai b and y = aj or y = aj b. Solution: Case 1: If x = ai and y = aj , the commutator is trivial. Case 2: If x = ai and y = aj b, then xyx−1 y −1 = ai aj ba−i aj b = ai aj ai baj b = ai aj ai a−j b2 = a2i , and thus each even power of a is a commutator.

For each x ∈ G we have λa (x) = ax and λa (ax) = a2 x = x, which implies that λa is a product of m transpositions (x, ax). Hence λa is an odd permutation since m is odd. Let H = {x ∈ G | φ(x) = λx is even}. Then H is a subgroup of G, and since a ∈ G − H, it is easy to check that [G : H] = 2, and so H is normal, contradicting the assumption that G is simple. REVIEW PROBLEMS 1. Prove that there are no simple groups of order 200. Solution: Suppose that |G| = 200 = 23 ·52 . The number of Sylow 5 subgroups must be a divisor of 8 and congruent to 1 modulo 5, so it can only be 1, and this gives us a proper nontrivial normal subgroup.