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

Q: Consider the group G = {x € R such that x + 0} under the binary operation x*y = -2xy The inverse…

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Q: Prove that any group with three elements must be isomorphic to Z3.

A: Let (G,*)={e,a,b}, be any three element group ,where e is identity.  Therefore we must have…

Q: Let m and n be relatively prime positive integers. Prove that the order of the element (1, 1) of the…

A: To prove ℤm×ℤn≅ℤmn ⇔ gcd(m,n)=1 Part 1) Proof of 'only if'  part  ℤm×ℤn≅ℤmn ⇒ℤm×ℤn is cyclic. Let…

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

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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…

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Q: Prove that the group G = [a, b

A: Given, the group G=a, b with the defining set of relations…

Q: Prove A3 is a cyclic group

A: We know that If G be a group of prime order then G is cyclic group

Q: What is the automorphism group of ?A ut (Z3, +)

A: Thanks for the question :)And your upvote will be really appreciable ;)

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

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Q: G be defined by f(r) = x1. Prove that f is operation-preserving if 6*. Let G be a group and f: G and…

A: To prove that the given function f is a homomorphism (operation preserving) if and only if G is…

Q: Prove that the trivial G is faithful if and only if G = {1G} representation of a nite group

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Q: Let G be a group with |G| = pq, where p and q are prime. Prove that every proper subgroup of G is…

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Q: integer 16. If EG→His a surjective homomorphism of groups and G is abelian, prove that H is abelian.…

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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: For any group elements a and x, prove that |xax-1| = |a|.

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Q: Show that if H is any group and h is an element of H, with h" = 1, then there is a unique…

A: Given that H is a group and h ∈H  Now,we define a mapping f:Z→H  such that   f(n) = hn for n∈Z For…

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

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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: let x be an element of group g. Prove that if |x|=n then x^-1=x^n-1

A: Given 'x' be an element of a group G and |x|=n. As G be a group , inverse of each element of G must…

Q: Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)-1 = a-'b-1 %3D

A: Given: The statement is, let a and b be elements of a group G. Prove that G is abelian if and only…

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: 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: 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: In Z, let A = <2> and B = <8>. Show that the group A/B is isomorphic to the group Z4 but…

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Q: In group theory (abstract algebra), is there a special name given either to the group, or the…

A: Yes, there is a special name given either to the group, or the elements themselves, if x2=e for all…

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

A: By definition (a-1)-1=a are both elements of a-1. Since in a group each element has a unique…

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

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Q: Let G be any group with the identity element e. With using the Group Homomorphism Fundamental…

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Q: If G is a cyclic group of order n, then G is isomorphic to Zn. true or false?

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Q: Let G be a group, and let xeG. How are o(x) and o(x) related? Prove your assertion

A: According to the given conditions:

Q: Use the three Sylow Theorems to prove that no group of order 45 is simple.

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

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Q: Find the group homomorphism between (Z, +) and (R- (0},.)

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Q: Let H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.

A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…

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: 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: (A) Prove that, every group of prime order is cyclic.

A: Let, G be a group of prime order. That is: |G|=p where p is a prime number.

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

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Q: Let G be an infinite cyclic group. Prove that G (Z,+)

A: To show that any infinite cyclic group is isomorphic to the additive group of integers

Q: Let G be a group such that a^2 = e for each aEG. Then G is * Non-abelian Cylic Finite Abelian

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

A: Concept:

Q: If A is an abelian group with A <G and B is any subgroup of G, prove that ANB < AB.

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Q: Find the order of each element of the group Z/12Z under addition

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Q: OLet a and b be elements of a group G. Prove that G is abelian if and only if (ab)- = a¯\b-!.

A: Prove that G is abelian if and only if (ab)-1=a-1b-1. For all a and b be elements of a group G.

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

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Q: Consider the group G = {x € R such that x # 0} under the binary operation *. ху X * y = x * 2 The…

A: First we have to find the identity element. Let G be the group and e be the identity element of G.…

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 that each non-zero element of an integral domain , regarded as a member of the additive group…

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