By Pierre Antoine Grillet

A thoroughly remodeled re-creation of this extraordinary textbook. This key paintings is geared to the wishes of the graduate pupil. It covers, with proofs, the standard significant branches of teams, earrings, fields, and modules. Its inclusive technique signifies that the entire worthwhile parts are explored, whereas the extent of element is perfect for the meant readership. The textual content attempts to advertise the conceptual knowing of algebra as an entire, doing so with a masterful grab of method. regardless of the summary material, the writer contains a cautious number of vital examples, including a close elaboration of the extra refined, summary theories.

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**Example text**

The canonical projection π : FX −→ X R is a homomorphism that extends ι to FX , since π ◦ η = ι ; therefore it is the homomorphism that 7. Presentations 33 extends ι to FX . If (u, v) ∈ R , then u v −1 ∈ N = Ker π , π (u) = π(v) , and (u, v) holds in X R via ι. Every element g of X R is the image under ϕ of an element a of F ; a is a product of elements of X and inverses of elements of X ; hence g is a product of elements of ϕ(X ) = ι(X ) and inverses of elements of ι(X ) . Definitions. X R is the (largest) group generated by X subject to every relation (u, v) ∈ R .

Show that every divisor of |Dn | is the order of a subgroup of Dn . 9. Find the order of every element of D4 . 10. List the elements of S4 and find their orders. 11. Show that the complex nth roots of unity constitute a cyclic group. Show that ωk = cos (2π k/n) +i sin (2π k/n) generates this cyclic group if and only if k and n are relatively prime (then ωk is a primitive nth root of unity). 12. Let A and B be subgroups of a finite group G . Show that |AB| = |A||B|/|A ∩ B| . 13. Find a group G with subgroups A and B such that AB is not a subgroup.

For example, the empty word and all one-letter words are reduced, for want of consecutive letters. If X = { x, y, z, . . }, then (x, y, z) and (x, x, x) are reduced, but (x, y, y , z) is not reduced. Reduction deletes subsequences (ai , ai ) until a reduced word is reached. 1 b when a = (a1 , a2 , . , an ) , ai+1 = ai , Definitions. In W , we write a −→ and b = (a1 , . , ai−1 , ai+2 , . , an ), for some 1 i < n ; k 1 1 1 1 b when k 0 and a −→ a −→ a −→ · · · −→ a (k) = b for we write a −→ some a , a , .