Let (A,) be a group. Let B be a subset of A such that 2|B| > |A|. Show that, for any a in A, a = b₁ b₂ for some by and b₂ in B. (One approach could be to consider set C = {a ★ b¹ | b € B})

Q: 1. Consider the group D5. Give five examples of D5 acting on a set, other than the trivial action.…

A: Given: Consider the group D5. Give five examples of D5 acting on a set, other than the trivial…

Q: Go Let G be a group and a, b = G. Prove that (ab)−¹ = b¯¹a-¹. Let G be a group. Prove that for all…

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Q: In the frieze group F7, write x-3zxyz in the form xnymz k.

A: It is known that F7=xnymzk∥n∈Z,m=0,k=0 or k=1 and x is the translation of R2, and it map u,v to…

Q: Define ∗ on ℚ+ by a ∗ b =ab/2 . Show that ⟨ℚ+,∗⟩ is a group.

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Q: In each case, find elements a and b from a group such that |a| = |b| = 2.   a. |ab| = 3 b. |ab| =…

A: a=2b=2

Q: Is the set Z a group under the operation a * b = a + b – 1? Justify your answer.

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Q: Let a be an element of a group and let |a| = 15. Compute the ordersof the following elements of G.a.…

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Q: Let a be a group element such that |a| = 48. For each part, find adivisor k of 48 such thata.…

A: Given: a be a group element such that a=48.We need to find divisors k of 48 such that the subgroups…

Q: (B) Let G be a set of all real numbers excep: . Define * on G by a * b = |a|b, where |a| is the…

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Q: Let the order of group G =8, show that G must have an element of order 2.

A: Let G be a group and O(G)=8 Also Let a∈G be an arbitrary element other than identity. Then the…

Q: Let a and b be elements of a group.  If |a| and |b| are relatively prime, show that intersects =…

A: Let m and n be the order of the elements a and b of a group G. Given that the orders of a and b are…

Q: In the frieze group F7, write x2yzxz in the form xnymz k.

A: It is given that, F7=xnymzk|n∈Z, m=0 or m=1, k=0 or k=1. we need to write x2yzxz  in the form of…

Q: Let G =<a, b | a2 =e, b2 =e, aba =bab>. To what familiar groupis G isomorphic?

A: Here given that  G =<a, b | a2 =e, b2 =e, aba =bab>. To what familiar group is G isomorphic…

Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: Let G, H, and K be groups. Show that (G x H) × K = G × (H × K). Hint: An element of (G × H) × K will…

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Q: Let (G, ★) and (G,o) be groups, with respective identity elements e and ē. Suppose there is a…

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Q: express Z10 as a product of Z5 x Z2 , verify that both groups are isomorphic.

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Q: Let a be an element of a group. Suppose abs(a5) = 12. Show that abs(a) = 12 or abs(a) = 60.

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Q: Determine whether the set G is a group under the operation * G= integers Z ; a*b= a-b

A: According to the given information it is required to check whether the given set G is a group or…

Q: 1. Let (G, *) and (H, *) be groups. The direct product of G and H is the set G x H equipped with…

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Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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Q: Let A and B be groups. Prove that A x B and B x A are isomorphic. Define a map from one set to the…

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Q: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

Q: Let G=.  How many elements does G have?  To what familiar group is G isomorphic?

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Q: Let G = <a, b |a2 = e, b2 = e, aba =bab>. To what familiar groupis G isomorphic?

A: It is given that, G = a, b : a2=e, b2=e, aba = bab be a group. We have to find the familiar group…

Q: Let G be a group and a E G be such that o(r) 20. Find the orders of r, xt, a0, al4 and r30. %3D

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Q: y" + 4y = 8 cos2x - 8e2x y(0) = -2 y'(0) = 0

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Q: Show that the frieze group F6 is isomorphic to Z ⊕ Z2.

A: Given That: A frieze group F6. To show that: The frieze group F6 is isomorphic to Z⊕Z2.

Q: let x be a non-empty set. show that whether (p(x).∩) is a group or not

A: Given: It is given that x be the non-empty set and to show that whether (p(x).n) is a group

Q: Define X = R \ {k} and define to be the operation such that x+y= = (x − k)(y − k) + k. Check each of…

A: The given set is X=ℝ-{k} and the operation * is given by x*y=(x-k)(y-k)+k. (X,*) will be called a…

Q: Find the outer set of points for group S.

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…