To which group is Z12/ ker f isomorphic to? (a) Z₁ (b) V4 (c) D4 (d) A₁

Q: We showed previously that Zp) = {|a.beZ and płb} is a subring of Q. %3D (a) Find (Zp)", the group of…

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Q: a b - Let H = 5, d E R, ad 0 Is H a normal sub- group of GL(2, R)?

A: Given: H=ab0d  a, b, d∈ℝ, ad≠0. We have to check whether H is a normal subgroup of GL2, ℝ or not.

Q: How does the subfield lattice of GF(230) compare with the subfieldlattice of GF(330)?

A: As per the question we have to compare the subfield lattice of the field GF(230) wity that of the…

Q: Let D10 = (a, b | a5 = b2 10. Assume x and y are two distinct elements of order 2 in D10. Let H =…

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Q: How does the subfield lattice of GF(230) compare with the subfieldlattice of GF(330)?

A: Consider the theorem "For each divisor m of n, GFpn, there exists a unique sub-field of order…

Q: Let F be a field, and consider the group G := 1 a b {6 {(₁; ;) = 01 с : a, b, c E F ≤ GL3 (F). : a,…

A: Center of group is set of all elements which commute with all elements of G

Q: Neutral element (Identity element) in the group {(Q\0),.} is   A 1   B -1   C Does not exist…

A: Given :- (Q\0) is a group under usual multiplication.  To find :- Neutral element (Identity element)…

Q: find the fundamental group of X := {(x, y, z) = R³|(x² + y²) (y² + z²)(x² + z² − 1) = 0}

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Q: Find the factor group G divides H where G is the quaternions,H=Z(G)

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Q: Let à = -4 (a) Find compab Answer= -15/9 Answer = 8 Answer = and b B (b) Find projab (i.e. the…

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Q: Q21 Let G={xER:x6=-1}. Define A on G by xAy=x+y+xy. (a) Show that gives a binary operation onG. (b)…

A: Introduction: Group is an algebraic structure in modern algebra. When a set along with a binary…

Q: Let G be the group with presentation (x,y: 1² = 1, r² = y²₁ xy = y ¹). Decide how many elements are…

A: Consider the given group G with presentation, x,y:x4=1, x2=y2, xy=yx-1 it can be observed that it is…

Q: Calculate G/H for G = V, the Klein’s four-group and H = (b).

A: The Klein four-group is defined by the group presentation V=a, b| a2=b2=ab=c2=e Given: G=e, a, b, c…

Q: (c) If K is a subgroup of G, then p(K) is a subgroup of H. Given: K < G (d) If K' is a subgroup of…

A: I have mentioned the test used to prove a subset to be a subgroup, as a note above. You can skip it…

Q: Let T: R² R² be given by T(x1, x₂) = -[at] It is known that I is an isomorphism. T-¹ is given by the…

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Q: There exists a quartic polynomial (i.e. a polynomial of degree 4) over Q whose Galois group is…

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Q: G is the group of nth roots of unity under complex multiplication Zn is the group of integers…

A: Consider the given information.

Q: — 2х21 Show that T: R² → R² defined by T()= - 15.) is an X2 isomorphism. X1

A: T is a n isomorphism

Q: 1. 4. Let a be a complex number such that lal > 2. Prove that 1. and a 1 generate a free group.

A: 4 Given abe a complex number such that a≥2. We have to prove that 10a1 and 1a01 generate a free…

Q: a b 2.3 Let e: (M,(R), +) → be defined by e a + d. where a,b,c,d ER (a) Prove that e is a…

A: Given that θ:M2ℝ,+→ℝ,+ is a mapping defined by θabcd=a+d where a, b, c, d∈ℝ. We have to prove that θ…

Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.

Q: Prove the general formula φ(gcd(a, b))φ(ab) = gcd(a, b)φ(a)φ(b).

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Q: (a) Obtain the splitting field of (x2 – 5) (x² – 7) over Q. Obtain the Galois group of this…

A: Find the attachment.

Q: 2.3 Let 9: (M2(R), +)→ be defined by e c d a b = a +d. where a,b,c,d €R (a) Prove that e is a…

A: A mapping f:(G,+)→(H,+) is said to be group homomorphism if it satisfies: f(a+b)=f(a)+f(b)…

Q: Let G = Z5OU(7). The identity of G is O (0, 1) O (1, 1) O None of the choices O (1, 0) O (0, 0)

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Q: Show that the determinant gives a group homomorphism det: GL(n,R)->R*, where R*=R-{0} under…

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Q: Q:: (A) Prove that 1. There is no simple group of order 200.

A: A simple group is a nontrivial group whose only normal subgroups are the trivial group and the group…