DEGiven the multiplication table below on (0,1,2,3,4,5,6,7},which statement best describes problem?هادات .......It is a non-abelian group.OJ123ellel-~34500 0 11م سرا -172 3 4 5 673 4 5 6 7062430 3 1 4215 41 3 22257It is not a group since it has no identity.50 6 75 0 6 72 3 4 1267456 6 4 177 34It is not a group since the operation is not associative.It is an abelian group.2 3 01 26It is not a group since the operation is not binary.7 0 67 550

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Answered: A D E Given the multiplication table…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 13E: Exercise 8 states that every subgroup of an abelian group is normal. Give an example of a nonabelian...

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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 .
Find all subgroups of the quaternion group.
The elements of the multiplicative group G of 33 permutation matrices are given in Exercise 35 of section 3.1. Find the order of each element of the group. (Sec. 3.1,35) A permutation matrix is a matrix that can be obtained from an identity matrix In by interchanging the rows one or more times (that is, by permuting the rows). For n=3 the permutation matrices are I3 and the five matrices. (Sec. 3.3,22c,32c, Sec. 3.4,5, Sec. 4.2,6) P1=[ 100001010 ] P2=[ 010100001 ] P3=[ 010001100 ] P4=[ 001010100 ] P5=[ 001100010 ] Given that G={ I3,P1,P2,P3,P4,P5 } is a group of order 6 with respect to matrix multiplication, write out a multiplication table for G.
Write 20 as the direct sum of two of its nontrivial subgroups.
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.
12. Find all normal subgroups of the quaternion group.
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.
Exercise 8 states that every subgroup of an abelian group is normal. Give an example of a nonabelian group for which every subgroup is normal. Exercise 8: Show that every subgroup of an abelian group is normal.

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