consider the permutation group SIdentify the elements of H from the list ofelements of S listed below by SAGEMATHsage: G=SymmetricGroup(4)sage: list(G)[O,(1,3)(2,4), (1,4)(2,3), (1,2)(3,4), (2,3,4),(1,3,2), (1,4,3), (1,2,4), (2,4,3), (1,3,4), (1,4,2),(1,2,3), (3,4), (1,3,2,4), (1,4,2,3), (1,2), (2,3),(1,3,4,2), (1,4), (1,2,4,3), (2,4), (1,3),(1,4,3,2),(1,2,3,4)]sage:

The Symmetric Group S4The symmetric group S4 is the group of all permutations of 4 elements. A…

Answered: consider the permutation group S…

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 2E

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Find two groups of order 6 that are not isomorphic.
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.
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.
Let Zeven represent the set of even integers, {... -6, -4, -2, 0, 2, 4, 6, ...}. Determine if the system (Zeven, +): (i) is closed; (ii) is associative; (iii) has an identity element; (iv) has inverses; and (v) is commutative. Is (Zeven, +) a semi-group? Explain Is (Zeven, +) a group? Explain
Are these sets with the given operation commutative groups? If not, indicate the reason. If several reasons apply, select the first reason that applies. ({0},+) ({-1,1},*) where a*b := a b (ℕ, +) ({1},*) where a*b := a b (ℤ3,*) where a*b := a b
Let f and g be permutations on the set {1, 2, 3, 4, 5, 6, 7}, defined as follows (b) f-1 (c) fgf = f= = 9 = Write each of the following permutations as a product of disjoint cycles, separated by commas (e.g. (1, 2), (3, 4, 5), … ). Do not include 1-cycles (e.g. (2)) in your answer. (a) fg= 1 2 3 4 5 6 7 24 6 5 1 3 7 1 3 2 3 4 5 6 1 2 7 5 6 4 Assume multiplication of permutations f.g obeys the rule (fg)(x) = f(g(x)) so (1, 3) (1, 2) = (1, 2, 3) not (1, 3, 2).
Let ?f be a permutation on the set {1,2,3,4,5,6}{1,2,3,4,5,6}, defined as follows
Determine if the following sets with operations are groups: (a) R with the operation given by r * y = -1 - y. (b) The subset of R® given by H = {(r,y, 2) € R* |x+ y+z = 0}, with the operation (x, y, 2) + (a, b, c) = (r +a, y +b, z + c).
Let Zodd represent the set of odd integers, {... -5, -3, -1, 1, 3, 5, ...}. Determine if the system (Zodd, +): (i) is closed; (ii) is associative; (iii) has an identity element; (iv) has inverses; and (v) is commutative. Is (Zodd, +) a semi-group? Explain Is (Zodd, +) a group? Explain
2. Let Zodd represent the set of odd integers, {... -5, -3, -1, 1, 3, 5, ...}. Determine if the system (Zodd, +): (i) is closed; (ii) is associative; (iii) has an identity element; (iv) has inverses; and (v) is commutative. Is (Zodd, +) a group? Explain Is (Zodd, +) a commutative group? Explain
Let ?f be a permutation on the set {1,2,3,4,5,6,7,8}{1,2,3,4,5,6,7,8}, defined as follows
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