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

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: Let G be a group. Using only the definition of a group, prove that for each a E G, its inverse is…

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Q: 1. Let G be an abelian group with the identity element e. If H = {x²|x € G} and K = {x € G|x² = e},…

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

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Q: (G, .) a group such that a.a = e for all a EG.Show that G is an abelian group. Let be

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Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…

A: We have given a map ,  ϕ : A → B , where A = ℝ , +   ,  B = ℝ* , · such that  , ϕx = 2x We know that…

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: 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: (H,*) is called a of (G,*) if (H,*) is a group.

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Q: 6. Let G be a group with the following property: "If a, b, and c belong to G and ab=ca, then b = c."…

A: For a group to be abelian it should follow that ab=ba for all a,b belonging to G. Now we will…

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: Show that Z is not isomorphic to Zž by showing that the first group has an element of order 4 but…

A: The given groups: Z5x and Z8x That is, Z5x =Number of elements of a group of Z5 Z8x =Number of…

Q: be the operation on Z defined by a*b = a+b for all a,beZ. Justify the following questions. 4 Let (1)…

A: Here * be the operation on ℤ defined by a*b=a+b4 for all a, b∈ℤ. We have to justify the followings:…

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: Let X be a group and we then let x and y be an element of X.  Prove that (x*y)^ -1 = a^-1 * b^-1 iff…

A: Since there are some mistakes in given typed question.question may like "Let X be a group and let…

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

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Q: Suppose that the fundamental group of X is Z and p(xo) is finite. Find the fundamental group of X.

A: Given fundamental group of X is  ℤ and p-1(xo) = finite value Now we have to find the fundamental…

Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…

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Q: G let then. [b, a]= be an group and Ta %3D

A: Given that G is a group and also a,b,c∈G. To prove that b,a= a,b-1 Since G is a group, it satisfies…

Q: Let G be a group, and let a E G.  Prove that C(a) = C(a-1).

A: Given: Let G be a group and let a∈G. then we will prove C(a)=C(a-1) If C(a) be the centralizer of a…

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: 1. Assume (X, o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) |x € X,y E Y} and define the…

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

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Q: Let ?1 , ?2 ??? ?3 be abelian groups. Prove that ?1 × ?2 × ?3 is an abelian group.

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Q: QUESTION 5 Show that ifevery element in a group G is equal to its own inverse, then G is Abelian.

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

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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: 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: If (G, * ) is a group with a a for all a in G then G is abelian

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Q: 1. State, with reasons, which of the following statements are true and which are false. (a) The…

A: Given Data: (a) The dihedral group D6 has exactly six subgroups of order 2. (b) If F is a free group…

Q: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

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: 6. Prove the following groups are not cyclic : (a) Z x Z (b) Z6 × Z (c) (Q+, ·) (Here, Q+ = {q € Q…

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Q: I'm unsure of how to approach this... Let N be a finite group and let H be a subgroup of N. If |H|…

A: According to the given information, Let N be a finite group and H be a subgroup of N.

Q: Assume that the equation zxy = e holds in a group. Then *

A: Given is  zxy = e  Thus, we can say  z(xy)  =  e  Let  xy = p  zp  =  e  And hence,  pz = e  =>…

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

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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: Construct the Cayley table for (Zo) ,c), and verify that this is an Abelian group.

A: 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 0 2 3 4 5 6 7 8 0 1 3 4 5 6 7 8 0 1 2 4 5 6 7 8 0 1 2 3…

Q: Let G = {a + b/2|a, b € Z}. Show that G is a group under ordinary addition.

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Q: Let G be a group and let x EG of order 23. Prove that there exist an element z EG such that z6 – x.

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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: Let x, y be elements in a group G. Prove that x^(−1). y^n. x = (x^(−1).yx)^n for all n ∈ Z.

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Q: Let c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…

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Q: Let G be a group, and assume that a and b are two elements of order 2 in G. If ab = ba, then what…

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Q: Let G be a group, a E G. Prove that a=a + a < 2

A: Concept:

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: 1. Consider the groups (R+, ) and (R,+). Then R* and R are isomorphic under the mapping $(x) = log10…

A: We use the definition of cosets, isomorphisms to answer these questions. The detailed answer well…