Which of the following groups are isomorphic? i) Z/75Z ii) Z/3Z ® Z/25Z iii) Z/15Z @ Z/5Z iv) Z/3Z @ Z/5Z® Z/5Z.

Q: Let G = Z8 x Z6, and consider the subgroups H = {(0, 0), (4, 0), (0, 3), (4,3)} and K = ((2, 2)).…

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Q: Give the lattice of subgroups of (Z2,), of (S,,0), and of (Z(20) ,)

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Q: Q 4 The inverse of the element (132) of the group S3 under function composi- tion o is (a) (12) (b)…

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Q: Consider the group G = Z4 X Z9 where the operation is done componentwise, that is (a, b) * (c,d) =…

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Q: The Galois group of  x4 +x2 +1 is the same as that  of  x6 -1 and is of order 2.

A: Since the primitive nth root of unity is        e2πi/n=cos2πn+i sin2πn. Then primitive third root…

Q: We will now consider three irreducible representations for the permutation group P(3): E A B I : (1)…

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Q: 12. Prove that the following groups are not cyclic: (a) Z2 x Z2 (b) Z2 x Z (c) Z x Z.

A: To show the following groups are not cyclic. If a group is cyclic the there exit an element in that…

Q: 3. You have already proved that GL(2, R) = {[ª la, b, c, d e R and ad – bc ± 0} forms a group under…

A: Note: There are two questions and I will answer the first question. So, please send the other…

Q: 16. Let G be a group and let g and h be in G. Show that . - as defined in #15.

A: Given: G is a group, g,h∈G, ϕg:G→G is a function defined as ϕg(x)=gxg-1 To prove: ϕg∘ϕh=ϕgh Let x∈G…

Q: Z, x is not a group.

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Q: 4. The group defined by the following table is called the group of quaternions. Use the table to…

A: NOTE:- According to the company guidelines we are supposed to answer only first 3 sub-parts. Kindly…

Q: . Let H (27) be the subgroup of G=Z36. What is the order of the quotient group G/H? = (a) 3 (b) 6…

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

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Q: 1. Consider H= (i) and K= (-1) the subgroups of Qg the quaternion group. 4) Explain why K is normal…

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Q: 17*. Find an explicit epimorphism from A5 onto a group of order 3

A: Epimorphism: A homomorphism which is surjective is called Epimorphism.

Q: is a group with identity (eg, eH).

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Q: 3. Use the three Sylow Theorems to prove that no group of order 45 is simple.

A: Simple group: A group G is said to be simple group if it has no proper normal subgroup Note : A…

Q: 4. a) Let H be the set of elements [a b] of G of GL(2,R) such that ab- bc = 1. Show that H is a…

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Q: Show that the group (U7, *) is isomorphic to {Z6, +}. how that there is an isomorphism between the…

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Q: 2. Suppose M and D are isomorphic groups. Prove that Aut(M) is isomorphic to Aut(D).

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Q: 14. Prove that the set of all rational number of the form 3"6" | m,nEZ} js a group under…

A: Denote with

Q: 3. Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the…

A: Result: Let G, * be a group and H is a subset. The subset H is said to be subgroup, if for every a,…

Q: 2. Find two distinct proper subgroups of the additive group (Z12, +}.

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Q: Let F = {f1; f2, f3, fa, f5. f6} where fi = x f2 = 1-x 1 f3 = fs = 1 fa fo under composition of…

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