#19 pls 19. Let L(G) be the set of normal subgroups of a group G. Showthat L(G) is a modular lattice. Show also that the lattice of allsubgroups need not be modular. [A lattice L is called modularif for all a, b, ce L, with a sc, av (b^ c) = (a v b)^c.]

Given: L(G) is the set of normal subgroups of a group G. We need to show, L(G) is a modular…

Answered: 19. Let L(G) be the set of normal…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 12E: 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...

See similar textbooks

Similar questions

Find two groups of order 6 that are not isomorphic.
Show that every subgroup of an abelian group is normal.
Find all homomorphic images of the quaternion group.
4. Prove that the special linear group is a normal subgroup of the general linear group .
9. Suppose that and are subgroups of the abelian group such that . Prove that .
5. Exercise of section shows that is a group under multiplication. a. List the elements of the subgroupof , and state its order. b. List the elements of the subgroupof , and state its order. Exercise 33 of section 3.1. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and is designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.
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.
Prove that the Cartesian product 24 is an abelian group with respect to the binary operation of addition as defined in Example 11. (Sec. 3.4,27b, Sec. 5.1,53,) Example 11. Consider the additive groups 2 and 4. To avoid any unnecessary confusion we write [ a ]2 and [ a ]4 to designate elements in 2 and 4, respectively. The Cartesian product of 2 and 4 can be expressed as 24={ ([ a ]2,[ b ]4)[ a ]22,[ b ]44 } Sec. 3.4,27b 27. Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 5.1,53 53. Rework Exercise 52 with the direct sum 24.
27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.
3. Consider the group under addition. List all the elements of the subgroup, and state its order.
32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping defined by is an automorphism of . Each of these automorphism is called an inner automorphism of . Prove that the set forms a normal subgroup of the group of all automorphism of . Exercise 20 of Section 3.5 20. For each in the group , define a mapping by . Prove that is an automorphism of .
Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.

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

19. Let L(G) be the set of normal subgroups of a group G. Show that L(G) is a modular lattice. Show also that the lattice of all subgroups need not be modular. [A lattice L is called modular if for all a, b, ce L, with a sc, av (b^ c) = (a v b) ^ c.]