12. For which integers n such that 3 ≤ n ≤ 11 is there only one group of order n (up to isomorphism) ?(A) For no such integer n(B) For 3, 5, 7, and 11 only(C) For 3, 5, 7, 9, and 11 only(D) For 4, 6, 8, and 10 only(E) For all such integers n

Given: 12. n is an integer such that 3≤n≤11. To find: For which value of n there is only one group…

Answered: 12. For which integers n such that 3 ≤…

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 18E

See similar textbooks

Similar questions

Find two groups of order 6 that are not isomorphic.
Let n be appositive integer, n1. Prove by induction that the set of transpositions (1,2),(1,3),...,(1,n) generates the entire group Sn.
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 .
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.
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?
Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units . Sec. 16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.
15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.
Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.
12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.
If a is an element of order m in a group G and ak=e, prove that m divides k.
42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .
10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .

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

12. For which integers n such that 3 ≤ n ≤ 11 is there only one group of order n (up to isomorphism) ? (A) For no such integer n (B) For 3, 5, 7, and 11 only (C) For 3, 5, 7, 9, and 11 only (D) For 4, 6, 8, and 10 only (E) For all such integers n