By W.J. Wickless

ISBN-10: 0824756274

ISBN-13: 9780824756277

Graduate textbooks frequently have a slightly daunting heft. So it truly is friendly for a textual content meant for first-year graduate scholars to be concise, and short adequate that on the finish of a path approximately the whole textual content could have been coated. This booklet manages that feat, solely with no sacrificing any fabric insurance. the traditional themes of staff thought, vector areas, modules, jewelry, box and Galois idea are coated, in addition to subject matters in noncommutative jewelry, crew extensions, and chosen issues in abelian teams. many of the brevity is because of the the truth that rather than supplying huge chunks of routines of middling trouble, Wickless (math, college of Connecticut) has opted to supply fewer routines of higher trouble.

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**Read e-book online A first graduate course in abstract algebra PDF**

Graduate textbooks frequently have a slightly daunting heft. So it really is friendly for a textual content meant for first-year graduate scholars to be concise, and short adequate that on the finish of a path approximately the full textual content may have been coated. This e-book manages that feat, solely with out sacrificing any fabric insurance.

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**Sample text**

A j-cycle ( k i , . . ,kj}. A transposition is a two cycle (a, b). , lr} = (f>. ,kj are imagined to be arranged in a circle, the reason for the terminology is clear. A j-cycle can be written in j different ways. For example, in 58, (2,5,3,7) = (5,3,7,2) = (3, 7,2,5) = (7,2, 5,3). 2 Let X,cr 6 Sn be disjoint cycles. Prove in at most three lines that ACT = a\. Here is an extremely useful result. 2 Every e ^ a 6 Sn can be written as the product of disjoint cycles. This representation is unique up to the order of the factors.

SYLOW THEOREMS 41 We present an interesting result, which will have applications in our next section. First, we need a preparatory lemma. , Nk be normal subgroups of a group G such that, for each fixed j, Nj n [Ni • • • ATj_! JVj+1 - • • Nk] —< e > . Then Ni---Nk \ — N I • • • Nk\. Proof. 10, the product of any finite collection of normal subgroups is a normal subgroup. The proof of our lemma is by induction on k, the case k — 1 being trivial. Suppose that k > 1 and that the result holds for any appropriate collection of m normal subgroups, whenever m < k.

10. DIRECT PRODUCTS 35 If all of the groups G; are additive, we call the Cartesian product with componentwise addition the (external) direct sum and adopt the notation 0™=1 Gi. The groups Gi are then called the summands of the direct sum. We caution the reader that, when dealing with a collection of groups {d : i £ I}, where the index set / is infinite, the notations Yli^i Gi and (J)ie/ Gi are defined differently. This situation will arise in several of the latter chapters. 1 Verify that H™=i ^ Jsa ' 9rouP- For all i, let €i e G; be the identity element.

### A first graduate course in abstract algebra by W.J. Wickless

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