2. In a group G, (ab) ³ = a³b³ for all a, bEG, and that 3 does not divide order ofG. Prove that every element in the group is cube of some element of G, i.e.,every element xEG is cube of some element yEG (Thus, x = y³).[Hint: It may be helpful to show that if a = b then a=b.)

Answered: Image /qna-images/answer/23ce6a8c-61a7-471c-a4e6-fe24bb322c96.jpg

Answered: 2. In a group G, (ab)³ = a³b² for all…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.8: Some Results On Finite Abelian Groups (optional)

Problem 15E: 15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that...

See similar textbooks

Similar questions

If a is an element of order m in a group G and ak=e, prove that m divides k.
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.
Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.
15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .
If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.
Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.
Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.
Construct a multiplication table for the group D5 of rigid motions of a regular pentagon with vertices 1,2,3,4,5.
Find two groups of order 6 that are not isomorphic.
24. Let be a group and its center. Prove or disprove that if is in, then and are in.
(See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup that leaves each of the elements 1,2,...,i fixed: Ki=gGg(k)=kfork=1,2,...,i For i=1,2,...,n. Prove that G=Sn if and only if HiHj for all pairs i,j such that ij and in1. A subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements of B there exists an element hH such that h(i)=j. Suppose G is a group that is transitive on 1,2,....,n, and let Hi be the subgroup of G that leaves i fixed: Hi=gGg(i)=i For i=1,2,...,n. Prove that G=nHi.

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

2. In a group G, (ab)³ = a³b² for all a, bEG, and that 3 does not divide order of G. Prove that every element in the group is cube of some element of G, i.e., every element xEG is cube of some element yEG (Thus, x = y³). [Hint: It may be helpful to show that if a³ = b' then a=b.)