34. Consider the action of S, the symmetric group of order 4, on Z[x1, X2, X3, X4] given by Let Hc S, denote the cyclic subgroup generated by (1423). Then, the cardinality of the orbit O,(x,x, +x,) of H on the polynomial x,x, + xx, is (a) 1 (c) 3 (b) 2 (d) 4

34. Consider the action of S, the symmetric group of order 4, on Z[x1, X2, X3, X4] given by Let Hc S, denote the cyclic subgroup generated by (1423). Then, the cardinality of the orbit O,(x,x, +x,) of H on the polynomial x,x, + xx, is (a) 1 (c) 3 (b) 2 (d) 4

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Show that SL(n, R) is a normal subgroup of GL(n, R). Further, by apply- ing Fundamental Theorem of…

A: Suppose, ϕ:GLn,R→R\0 such that ϕA=A for all A∈GLn,R Now, sinceA∈GLn,R if and only if A≠0 Now, we see…

Q: true O false O * -2 2) Let |G| = n and H a subgroup of G then la, + H| = la, + H| = laz * H| = ... =…

Q: If (G,X) be the group of all real 2 x 2 matrices (b) such that ad - bc #0 with matrix multiplication…

A: Given if G, × be the group of all 2×2 matrices abcd such that ad - dc≠0 with matrix multiplication…

Q: Let G 0 -{(81%) = : a, b ER, a 0 (a) Show that G is a subgroup of SL(2). (b) For each of the…

A: Since you have posted a question with multiple subparts, we provide the solution only to the first…

Q: If n be any given positive integer, show that the mapping f:C\index{0}\rightarrow{}C\index{0}…

Q: Suppose z is implicitly given as a function of x and y, according to The equation: y−zx3=1−yz2 The…

A: Suppose z is implicitly given as a function of x and y, according to The equation: y−zx3=1−yz2 The…

Q: Prove that {(1), (1, 2)(3, 4), (1, 3)(2,4), (1, 4)(2,3)} is a subgroup of S4.

A: We use the Cayley table.

Q: 3. (a) (i) Let p: (Z, +) → (C – {0}, :) be given by (n) = (-i)", for each n e Z. Prove or disprove…

Q: 1.) Let G₁ = £1₁ 1₁ i -i) be group of 4 Complex numbers under mutiplication. a) is £1, i} a subgroup…

A: Given group under multiplicationG = 1, -1, i , -i A subset H of G will be subgroup of G if H…

Q: What is the automorphism group of ?A ut (Z3, +)

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Show that PSL(2, Z7) = SL(2, Z7)/Z(SL(2, Z7)), which has order 168,is a simple group.

A: Given: PSL2, Z7=SL2, Z7/ZSL2, Z7 has order 168.To show that PSL2, Z7 is a simple group for that we…

Q: 8) a) Let A be the set of all real numbers z 0, 1, 2. Define the permutation f: A→A by the formula…

Q: Q/ In (Z6 , +6 ) find the cyclic subgroup generated by 1, 2, 5.

Q: Define on R the operation * by x*y = X+y+k, for all x,y element of R and k is fixed real number. The…

A: We have to check

Q: Prove that a group of class equation 24 = 1+1+4+4+4+4+6 does not have a normal subgroup of order 4.

A: Prove that a group of class equation 24 = 1+1+4+4+4+4+6 does not have a normal subgroup of order 4.

Q: Consider the group G = ℚ* × ℤ with operation * on G that can be expressed as: (w, x) * (y, z) =…

Q: In Z₁6, write down all the cosets of the subgroup H = {[0], [4], [8], [12]}. (Since the operation in…

A: We see, elements of Z16 are {[0], [1], [2], [3], [4], [5], [6], . . . . . . . , [15]} H = {[0], [4],…

Q: Determine the Galois group of x2 +9 over R.

A: The objective of the question is to determine the Galois group of x2+9 over R.

Q: 3. In ROR under componentwise addition, let H = {(r,3r): E R}. (a) Show that H is subgroup of ROR.…

Q: [S,*]= [Z,+] and [S',*']=[Z,∘] where a∘b=a+b-1. Then ?(n)=n+1 is an isomorphism from [Z,+] to [Z,∘]

Q: 2 Prove That D₂ = 8 cannot be the internal direct product of two of its proper subgroups.

Q: Show that the Galois group of a polynomial of degree n has orderdividing n!

A: Let us consider that the polynomial, fx=∑i=0ncixin, ci∈F Also let K be the splitting field of fx…

Q: Prove that x9 - 1 and x7 - 1 have isomorphic Galois groups over Q.

Q: chow that An s a Group with respect, to Cooplesition of functjon.

Q: G = (R, +), H = {a+bv2: a,b € Z}

A: Given G = (R, +), H = {a+b√2 : a, b∈Z}. We check whether H is a subgroup of G.

Q: 1. Let 0(V) be the set of all orthogonal transformations on V. Prov O) is a group with respect to…

A: Let O(V) be the set of all orthogonal transformations on V. The determinant of an orthogonal matrix…

Q: Show that in C* the subgroup generated by i is isomorphic to Z4.

A: C* is group of non-zero comples numbers with multiplication

Q: Let Ø: Z50 → Z15 be a group homomorphism with Ø(x) = 4x. Ø-1(4) None of the choices O {1, 16, 31,…

Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D where a = (1 23 4) and b…

Q: 5. Prove (using the subgroup condition rather than using the axioms) that the set a b H = {( 6 ) :…

Q: Consider the group G = ℚ* × ℤ with operation * on G that can be expressed as: (w, x) * (y, z) =…

Q: Let f.g:(G,.) (G')and h(m) f(m). g(m), Vm € G, them h is group homomorphism if f is homomorphism O f…

Q: f: (A, SA)(B,S) be an order isomorphism and, Sa={xEA: x<naf## and Spa) =y eB= y <gf(@)}. 3) Let %3D…

A: Consider the given information: Let f:A,≤A⇒B,≤B be an order isomorphism. Sa=x∈A: x<Aa≠∅ and…

Q: Q/ Let G = Z and a * b = a + b show that (G,* ) is a group or not.

Q: (Z., +6) is a subgroup of (Zg, +8)

Q: 3. Let D be an integral domain. (a) Prove that FD is an abelian group under the operation of…

A: Let us prove that field of fractions is an abeliean group under addition.

Question

34. Consider the action of S, the symmetric group of order
4, on Z[x,, x2, X3, X4] given by
Oe S,.
Let Hc S, denote the cyclic subgroup generated by
(1423). Then, the cardinality of the orbit O,(x,x3+xx)
of H on the polynomial x,x, + x,X4 is
(a) 1
(c) 3
(b) 2
(d) 4

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Groups$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Q2) Consider the subgroup H = (i) of the the group (C\{0},.). (1) Find H (2) How many element of order 2 in H? explain that H is normal subgroup of (C\{0},.). (3) Show
Which of the following is not an automorphism? Ø (R',. ) » (R ) with Ø(x) = In x O None of the choices O This Option ở: (Z, +) - (Z. +) with 0(x) 3r This Option This Option
8. Let G = U= {z = C||z| = 1} be the circle group. Then X = C, the set of complex numbers, is a G-set with group action given by complex number multiplication. That is, if z € U and w € C, *(z, w) = zw. Find all the orbits of this action. Also, find XG.
Q1\Find the inverse for each element in the following mathematical systems (Z, *) where * defined as a*b= a+b +11 Va, b €Z. The Kline four group. A) B) C) (D) (E) (Ze, +). (R, *) where * defined as a*b= a+b+ab Va, b ER. (P(X), A) where X-(0, 1, 2). dlic groun has a unique ident
Prove (using the subgroup condition rather than using the axioms) that the set a b H 1 = {(88): Q,CER, DER} 0 с is a subgroup of GL2(R)
Let H be any complexes of a group G, then (HK)\power{-1}=K\power{-1}H\power{-1}
Prove that that A6 has no subgroup of order 120.
8. Let the set G ZXZ and the binary operation on G be given by the rule (a, 6) • (c, d) = (a + r, 6+ d). It is easily verified that the pair (G, ) is a commutative group. a) Show that the mapping f: G- Z defined by fl(a, b)] = a is a homomorphism from (G, ) onto the group (2,+). b) Determine the kernel of this mapping. c) If II = {(a, a) | a E Z}, prove that (H, ) is a subgroup of (G, ), which is isomorphic to (Z,+) under the function f.
Let (IR,+) be a group of real numbers under addition and (R+,-) be the group of positive real numbers under multiplication. Prove f: R→ R+ by f (x)= ex for all x ER is homomorphism and isomorphism.
How do you interprete the main theorem of Galois Thoery in terms of subgroup and subfield diagrams?
Consider the following map defined on the symmetric group S3: Þ: S3 → GL₂(R) given by: Þ(e) = Þ((123)) = Þ((132)) = (19) ((12)) = ((13)) =((23))= -29 20 -42 29 (a) Show that map & defines a representation of S3.
9. a) find two integers x,y such that 30x+101y=1. (hint: gcd(30,101)=1). b) find the inverse of 30 in the group U(101).