I. Provide a two-column proof to the following statements.1. Let G be a group with the property that for any x,y, z in the group, xy = zximplies y = z. Prove that G is Abelian.2. Prove that a group G is Abelian if and only if (gh)-1 = g¬1h-1 for all g,h e G.3. Prove that in a group, (g¬')-1 = g for all g.

Answered: Image /qna-images/answer/5522547a-ab3e-4c65-8c56-22d596a9b3ce.jpg

Answered: I. Provide a two-column proof to the…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.1: Definition Of A Group

Problem 5TFE

See similar textbooks

Similar questions

Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)
Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.
If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.
Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.
24. Let be a group and its center. Prove or disprove that if is in, then and are in.
Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.
Label each of the following statements as either true or false. Let x,y, and z be elements of a group G. Then (xyz)1=x1y1z1.
9. Suppose that and are subgroups of the abelian group such that . Prove that .
If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.
For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication. (Sec. 3.5,3,6, Sec. 4.6,17). Find an isomorphism from the additive group 4={ [ 0 ]4,[ 1 ]4,[ 2 ]4,[ 3 ]4 } to the multiplicative group of units U5={ [ 1 ]5,[ 2 ]5,[ 3 ]5,[ 4 ]5 }5. Find an isomorphism from the additive group 6={ [ a ]6 } to the multiplicative group of units U7={ [ a ]77[ a ]7[ 0 ]7 }. Repeat Exercise 14 where G is the multiplicative group of units U20 and G is the cyclic group of order 4. That is, G={ [ 1 ],[ 3 ],[ 7 ],[ 9 ],[ 11 ],[ 13 ],[ 17 ],[ 19 ] }, G= a =e,a,a2,a3 Define :GG by ([ 1 ])=([ 11 ])=e ([ 3 ])=([ 13 ])=a ([ 9 ])=([ 19 ])=a2 ([ 7 ])=([ 17 ])=a3.
15. Prove that if for all in the group , then is abelian.
Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS