Let X be a set and define the binary operation ∆ on the subsets of X by A∆B := (A∩Bc)∪(B∩Ac). Prove that the set P(X) of all subsets of X is a group under the operation ∆.

Q: Let G be a group of order pm where p is a prime number and m is a positive integer. Show that G…

A: Let G be a group of order pm, where p is a prime number and m is a positive integer. Then,…

Q: Prove that the set of all orientation-preserving isometries of R³ is a group.

A: We know that the identity transformation i is an orientation-preserving and also an isometrics as…

Q: Let G be a and group Z(G)=< g€G•xg=gx, VX€G} %3D be the center of G. Show that G is commutative if…

A: Solve the following

Q: If Φ is a homomorphism from Z30 onto a group of order 5, determinethe kernel of Φ.

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Q: Prove that a group of order 12 must have an element of order 2.

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Q: is] Let G and H be groups, and let T:G→H_be Isomorphism. Show that if G is abelian then H is also…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Consider the group G = {x € R such that x* 0} under the binary operation x*y=-

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Q: Let (G,*) be a semigroup with identity e e G. Show that H = {a E G/a invertible} C G is a…

A: A set R is called a group with a binary operations * if it satisfies the given below conditions for…

Q: Let G be a group. Show that for all a,b,c G, (acb-1)-1 = bc-la.

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Q: For a group G and a fixed element a ∈ G, define the subset C(a) to be the set of all elements in G…

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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…

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Q: Let U(n) be the group of units in Zn. If n > 2, prove that there is an element k EU (n) such that k2…

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

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Q: 3.) Prove that for every element a of a Group G, Z(G) is a subset of C(a)

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Q: Prove that a group G is abelian if and only if (ab)-1 = a-lb¬ va,bEG

A: We need to prove that a group G is abelian if and only if (ab)-1=a-1b-1 , for all a,b in G. The…

Q: In the set of real numbers R there is an operation defined as: x x y = Vx³+y³ prove that (R, x) is…

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Q: Let G be a finite cyclic group of order n with generators a and b . Prove that the mapping ϕ(ai)=bi…

A: Given that, G is a finite cyclic group of order n with generators a and b. Also, the mapping is…

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

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Q: Let (G, *), (G', *' and (G", ") be groups, and let :G G' and : G' → G" be isomorphisms. Prove that t…

A: We will use the definition of isomorphism to prove that the composition of two isomorphisms is again…

Q: If f: G to H is a surjective homomorphism of groups and G is abelian, prove that H is abelian.

A: As we know that a group homomorphism f:G to H is a map from G to H satisfying:

Q: Prove that in a group, (a-1)¯' = a for all a.

A: To prove that in a group (a-1 )-1=a for all a.

Q: Prove the statement: Let g be a group, if g is abelian then (ab)2 =  a2b2.

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Q: Let H be the set of all elements of the abelian group G that have finite order. Prove that H is a…

A: Let H be the set of all elements of the abelian group G that have finite order.  Prove that H is a…

Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

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Q: Prove that each of the following sets, with the indicated operation, is an abelian group. (a) (R, *)…

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Q: For each element a in the group G, define the mapping ka: G → G by ka(x) = xa-'for all x in G. %3D…

A: Since, you have posted multiple questions, but according to guidelines we will solve first question…

Q: (b) Given two groups (G,) and (H, +). Suppose that is a homomorphism of G onto H. For BH and A:{g…

A: Given that G,·and H,* are two groups.

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

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Q: Let G be a group and define the map o : G → G by $(g) = g¬1. Show that o is an automorphism if and…

A: Automorphism and abelian group

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

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Q: Determine whether an onto homomorphism between the groups D6 and D3 + Z2 exists.

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Q: Let (G, ) be a group. Define a new binary operation * on G by the formula a * b = b · a for all a, b…

A: We proved (G,*) is a group if it satisfied the following axioms.

Q: If G is a cyclic group of order n, prove that for every element a in G,an = e.

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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: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

A: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

Q: Let G be an abelian group, then (acba)(abc)¯1 is

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Q: Let (G,*) and (H,*) be finite abelian groups. If G x G = H x H then G=H. Show that they are…

A: Given that, G×G=H×H⇒G=H Since G,* and H,* are both finite abelian groups we get,…

Q: Prove that the bijective functions of a set A in it, with the operation composition form a group.

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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: For each binary operation * defined on a set below, determine whether or not * gives a group…

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Q: Q- Let S be a set on which an associative binary operation · has been defined such that S contains…

A: To verify that U(S) satisfies the axioms of a group (under the given conditions)

Q: Let (G1, ) and (G2, *) be two groups and p: G1 G2 be an isomorphism. The

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Q: Let G be a group and let p: G → G be the map p(x) = x-1. (a) Prove that p is bijective. (b) Prove…

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Q: In the set of real numbers IR there is an operation defined as: I x y= Vr³+y³ %3D prove that (IR, ×)…

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Q: / Let x be anan- empty set, show that whe ther (P(X), n) is a a group of not

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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:…

A: The answer is given as follows :

Q: Let G and H be groups, and let ø:G-> H be a group homomorphism. For xeG, prove that )x).

A: Given:G and H be the groups.

Q: Verify that (ℤ, ⨀) is an infinite group, where ℤ is the set of integers and the binary operator ⨀ is…

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.