# Download PDF by David M. Burton: Introduction to Modern Abstract Algebra

By David M. Burton

ISBN-10: 0201007223

ISBN-13: 9780201007220

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Example text

9. Let a and b be intep;erR, not both zero. The least common multiple of a and b, denoted by lem (a, b), iH the positive integer e such that I I 1) a e and b e; that iH, e is a multiJlle of both 2) if a I e and b I c, tht'n e I c. a and b, Show that the least common multiple of a and b is related to the greatest common divisor of a and b by (11:m (a, b) )(gl:d (a, b» == labl. 20. Let a, b, c E Z, with a and b not hoth zero, and let d = ged (a, b). Verify that there exiMt intcgcrM x and 1/ Hueh that ax if and only if die.

For the proof of the next theorem, we shall require a prl'liminary lemma. Lemma. If a, b, c, dE G and (G, *) is a 8('migroup, then = (a * b) * (c * d) a * «b * c) * d). Proof. (! lw produet c * d by x. , WI! hnvn a * «b * c) Then, Killce the * d) = a * (b * (c * d» = a * (b * x) (a * b) * x = (a * b) * (c * d). Theorem 2-3. If (G, *) is a group and a, bEG, then (a * b)-J = b- I * a-I. That is, the inv('rRC of a prodm·t of group clements is the product of their inverses in reverse ordl·r. Proof.

The subsequent lemma will serve to isolatc the most tedious aspect of the theorem. Lemma. If a and bare noncom muting elements of a group (G, *)-that is, a * b ¢ b * a-then the clements of the set, {e, a, b, a * b, b * a}, are all distinct. Proof. The basic idea of the proof is to examine the members of the set {e, a, b, a * b, b * a} two at a time, and show that each of the ten possible equalities leads to a contradiction of the hypothesis a*b¢b*a. On several occasions, the cancellation law is used without explicit reference.