Show that Z12 is not isomorphic to Z2 ⊕ Z6.  ℤn denotes the abelian cyclic group of order n. Justify your answer in detail.

Q: a. Show that (Q\{0}, * ) is an abelian (commutative) group where * is defined as a ·b a * b = .

A: To show this we have to show that this holds closure, associative, identity, inverse and commutative…

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

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Q: Prove that GL(2,R) is not an abelian group

A: Solution is given below:

Q: (1) Z/12Z (2) (Zx 끄)/(6Zx 14Z) (3) (Z4 × Z4)/((2, 3)) (4) (Z4 x Z10)/((2, 4))

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Q: 3. Prove that (Z/7Z)* is a cyclic group by finding a generator.

A: Using trial and error method, seek for an element of order 6.

Q: 2- Let (C\{0},.) be the group of non-zero -complex number and let H = { 1,-1, i,-i} prove that (H,.)…

A: To Determine: prove that H,. is a subgroup of a group of non zero complex number under…

Q: Let G be a group containing elements a and b. Express (ab)^2 without parentheses. Do not assume that…

A: Given that G is a group and let a,b belongs to G. The given expression is

Q: prove that the group G=[a b] with defining set of relations a^3=e, b^7=e, a^-1ba=b^8 , is a cyclic…

A: We need to prove that , group G = a , b  with defining sets of relations a3 = e   ,  b7 = e  also…

Q: Consider the square X = [-1,1]2 = {(x, y)|x > -1, y < 1} and 0 = (0,0). Show that the fundamental…

A: image is attached

Q: If x is an element of a cyclic group of order 15 and exactly two of x3, x5, and x9 are equal,…

A: Given: The order of group is 15

Q: Prove or Disprove that the Klein 4-group Va is isomorphic to Z4.

A: The statement is wrong.

Q: Let G be a group. Show that for all a, b E G, (ab)2 = a2b2 G is abelian

A: To prove that the group G is commutative (abelian) under the given conditions

Q: The group GLQ,R) abelian group is an

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Q: Let G be a group and let r, y e G such that ya = r-ly. Use the Principle of Mathematical Induction…

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Q: Prove that a group of order 3 must be cyclic.

A: Given the order of the group is 3, we have to prove this is a cyclic group.

Q: Let G be a group. V a,b,c d and x in G, if axb=cxd then ab=cd then G is necessarily: * O Abelian O…

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Q: Describe all the elements in the cyclic subgroup of GL(2,R) generated by the given 2 × 2 matrix. -1

A: Let A=0-1-10∈GL2, R Then the cyclic subgroup generated by A denoted by A=An| n∈Z

Q: Let S = R\{-1} and define a binary operation on S by a*b = a + b + ab. Prove that (S, *) is an…

A: 2) S=R∖-1 binary operation defined by a*b=a+b+ab

Q: LetS=R{−1} and define a binary operationon S  by a∗b=a+b+ab. Prove that (S, ∗) is an abelian group.

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Q: find Aut(Z30).  Use the FUndamental Theorem of Abelian Groups to express this group as an external…

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Q: Prove that the group G with generators x, y, z and relations z' = z?, x² = x², y* = y? has order 1.

A: In order to solve this question we need to make the set of group G by finding x, y and z.

Q: Prove that if a is the only element of order 2 in a group, then a lies inthe center of the group.

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Q: If f (x) is a cubic irreducible polynomial over Z3, prove that either xor 2x is a generator for the…

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Q: Use the Cayley table of the dihedral group D3 to determine the left AND right cosets of H={R0,F}.…

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Q: Define a binary operation on R\ {0}by x*y = . 2 Prove that this set with this binary operation is an…

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Q: Q3: (A) Prove that 1. There is no simple group of order 200.

A: Simple group of order 200

Q: Consider the group G = {x € R]x # 1} under the binary operation : *• y = xy – x-y +2 The identity…

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Q: Prove that every finite Abelian group can be expressed as the (external) direct product of cyclic…

A: Fundamental Theorem of Finite Abelian Groups: Every finite Abelian group is a direct product of…

Q: (а) (Q, +) (b) (Zs, ·)

A: (a)(ℚ,+)(b)(ℤ8,.)

Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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Q: not

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Q: Use the left regular representation of the quaternion group Q8 to produce two elements of Sg which…

A: Fix the labelling of Q8 , Take elements 1, 2, 3, 4,  5, 6, 7, 8 are 1, -1, i, -i, j, -j, k, -k…

Q: Let S = R\{-1}. Define * on S by a * b = a+b+ ab. Prove that (S, *) is an abelian group.

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Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

A:

Q: If I is an ideal of R, then by definition, (I, +) is an abelian group. Consequently, it has an…

A: Since I is ideal , therefore I is definitely a subset of the ring R.

Q: 5. Prove that the cyclic group Z/15Z is isomorphic to the product group Z/3Z × Z/5Z.

A: Definitions: Isomorphism: A mapping between two sets is called an isomorphism if it is one-to-one,…

Q: If G is a non-abelian group of order 8 with Z(G) {e}, prove that |Z(G)| = 2.

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Q: Let G be a group. V a, b, c d and x in G, if axb cxd then ab = cd then G is necessa: Abelian Of…

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Q: Decide if the abelian group Z/2 × Z/2 is cyclic or not. Prove your answer

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Q: If R2 is the plane considered as an (additive) abelian group, show that any line L through the L in…

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Q: Let G be a group. V a,b,c d and x in G, if axb=cxd then ab=cd then G is necessarily: O Abelian O Of…

A: Solution:Given    G be a group∀a,b,c,d and x in G

Q: show that under complex multiplication, G={1,-1.i,-i} is an abelian group?

A: we have proved this by cayley table.

Q: (d) Prove that 1+ N is a group with respect to multiplication. (e) Verify that 1+N(Z27) is a cyclic…

A: d) To prove that 1+N is a group. Let a=1+n, b=1+n'∈1+N where n,n'∈N=Nℤn. Then,…

Q: 22. Prove that the set = {(₁ ~ ) 1} x) | : x, y ≤ R, x² + y² = 1 = SO(2) = forms an abelian group…

A: Given: 22. SO(2)=x-yyx : x, y∈ℝ, x2+y2=1 To show: The given set is a group with respect to…

Q: Every abelian cy elic O True O False group is

A: to check  whether every abelian group is cyclic or not? proof let a euler group U8=1,3,5,7 let…

Q: On Z is defined the following binary operation; x . y = x + y - 1 Show that (Z, . ) is an abelian…

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Q: Show that the translations of Rn form a group.

A: we have to show that   the translations of Rn form a group   We know that

Q: Show if all primitive transformations of the nonzero form x '= x ,y' = cx + dy d are a group.

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Q: Prove that the symmetric group (S₂, 0) is abelian.

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Q: Use the fundamental theorem of Abelian groups to express Z20 as an external direct product of cyclic…

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