1. Let G.*) be a group and a EG Suppose that a*a = a Prove or disprove that a must be the identity element.

Q: Consider the group (Z,*) defined as a*b=a=b , then identity (Neutral) element is

A: Given that ℤ,* is a group. where * is defined as a*b=a=b. That is a-b=0. To find the neutral element…

Q: 1. Let a and b be elements of a group G. Prove that if a E, then C. 2. Let a and b be elements of a…

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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: If g and h are elements from a group, prove that ΦgΦh = Φgh

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Q: Let G be a group and let a,b element of G such that (a^3)b = ba. If |a| = 4 and |b| = 2, what is…

A: see below the answer

Q: 6. Show that for any two elements x, y of any group G, o(xy) = o(yx). %3D

A: Fact 1:In a group G, if x∈G such that xn = e, then O(x)|n (e→ identity element.)i.e. order of x…

Q: Let G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that…

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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: Show that group U(1) is isomorphic to grop SO(2)

A: The solution is given as follows

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

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Q: If a is a group element, prove that every element in cl(a) has thesame order as a.

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Q: Prove that E(n) = {(A, ¤) : A e O(n) and E R"} is a group. %3D

A: Consider the given: E(n)={(A,x)} where A∈O(n)and x∈ℝn

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

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

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Q: Show that each of the following is not a group. 1. * defined on Z by a*b = |a+b|

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Q: Let C be a group with |C| = 44. Prove that C must contain an element of order 2.

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Q: For any elements a and b from a group and any integer n, prove that(a-1ba)n = a-1bna.

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Q: Show that if a group has an even number of elements then there is a element A other than unity such…

A: We have to prove that if a group has an even number of elements then there is aelement A other than…

Q: a The group is isomorphic to what familiar group? What if Z is replaced by R?

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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: W6 Assume that H, k, and k are SubgrouPs of the group G and k, , Ka 4 G. if HA k, = HN k Prove that…

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Q: c) Show that Z,,+, is a cyclic group generated by 3

A: 3(c) To check if 3 is generator of (Z5 , +5) , we must check that 3 generates all the members of Z5…

Q: What are the three things we need to show to prove that an ordered pair is a group?

A: We have to give the properties of an ordered pair to prove that it is a group.

Q: Let G be a group and let a e G. In the special case when A= {a},we write Cda) instead of CG({a}) for…

A: Consider the provided question, According to you we have to solve only question (3). (3)

Q: 3. Consider the group (Z,*) where a * b = a + b – 1. Is this group cyclic?

A: 3. Given the group ℤ,* where a*b=a+b-1. Then, 1*x=x*1=x+1-1=x Here 1 serves as the identity for Z.

Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, Cg(A) = A and…

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Q: 1- Prove that if (Q - {0},) is a group, and H = 1+2n 1+2m 9 n, m e Z} is a subset of Q-{0}, then…

A: We need to prove that H=1+2n1+2m∋n,m∈Z is a subgroup of Q-0 A subset W of a group V is said to be a…

Q: Show that the groups (Z/4, +4) and (Z/5 – {[0]}, x5) are isomorphic.

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Q: Explain the following statement "If G is a group an a E G then o(a) = | |." 31. %3D

A: Order of element a ∈ G is the smallest positive integer n, such that an= e, where e denotes the…

Q: If (G, ) is a group with a = a for all a in G then G is %D abelian

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Q: Suppose G is a group and r, be G so that r = b and r = b. Solve for a in terms of b.

A: Given: G is a group, and x,b∈G, so that x3=b5 and x8=b2. Formula used: Basic formula in power and…

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: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({3* : k E…

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Q: belong to a group. If |a| = 12, |b| = 22, and (a) N (b) # {e}, prove that a® = b'1.

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Q: Prove or disprove, as appropriate: If G x H is a cyclic group then G and H are cyclic groups.

A: GIven two groups G and H such that GxH is cyclic. True or false: G and H themselves are cyclic

Q: Let G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

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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…

Q: Let G be the set (te, ±a, ±b, ±c) where [ 1 10 Show that G form a Group under multiplication.

A: Group: If a set (finite or infinite) together with a binary operation satisfies the four basic…

Q: If G is a group with identity e and a2 = e for all a ∈ G, then prove that G is abelian.

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Q: Let a and b be elements in a group G. Prove that ab^(n)a^(−1) = (aba^(−1))^n for n ∈ Z.

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Q: Show that if p and q are distinct primes, then the group ℤp × ℤq is isomorphic to the cyclic group…

A: We have to show that if p and q are distinct primes, then the group Zp×Zq is isomorphic to the…

Q: Define * on Q by a +b= qb Is Q a group under *? 210

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Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, CG(A) = A and…

A: Given G=S3 and A=1,1 2 3, 1 3 2 The objective is to show that CGA=A, NGA=G. The definition for…

Q: G, ba = ca implies b = c and ab = ac implies b = c for elements a, b, c E G. 31. Show that if a? = e…

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Q: 25. Prove that R* x R is a group under the operation defined by (a, b) * (c, d) = (ac, be + d).

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Q: 4. Consider the additive group Z. Z Prove that nZ Zn for any neZ+.

A: We know that a group G is said to a cyclic group if there exists an element x of the group G such…

Q: Consider the set of permutations V = {(1), (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)}. Determine whether…

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Q: 64

A: Under the given conditions, to show that the cyclic groups generated by a and b have only common…