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

Q: Let (G1, ) and (G2, *) be two groups and p: G1- G2 be an isomorphism. Then * G2 is abelian if G1 is…

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Q: Prove that a group of order 175 is Abelian.

A: Let G be a group of order 175 We first try to rewrite 175 as prime factorization as follows: 175 =…

Q: Q3\ Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A: (G, *) be a finite group of prime order   To prove (G, *) is an abelian group

Q: (c) Prove that if G is a (not necessarily abelian) group, a, b e G, and a² = b² = (ab)² = e, then ab…

A: Use property of group and solve it.

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Q: 6. Let : (G,*) → (G',') be a group isomorphism, and let a € G. Prove that (a). G is abelian if and…

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Q: (a) Show that a group G is abelian, if (ab)² = a²b², for a, b € G[C.

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Q: Let (G1, •) and (G2 , *) be two groups and :G1 G2 be an isomorphism. Then * G2 might be abelian even…

A: we know that,  If two groups are isomorphic, then all the property will be similar i.e. one  is…

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Q: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group {e, a, b, a…

A: Given :              A group of order 4  Corollary :                    The order of an element of a…

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: QUESTION 6 Prove that a cyclic group G= is Abelian.

A: Given cyclic group G = <a>

Q: Let (G1, •) and (G2, *) be two groups and p: G1→ G2 be an isomorphism. Then * G2 might be abelian…

A:

Q: If G is a finite group and some element of G has order equal to the size of G, we ca say that G is:…

A: an abelian group, also called a commutative group, is a group in which the result of applying the…

Q: Q3\Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A:

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: 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: Let (G1, ) and (G2, *) be two groups and o: G1 G2 be an isomorphism. The O G2 might not be abelian…

A: Two group are isomorphic the the properties of groups are must be same, i.e,  if one group is an…

Q: Prove that if (ab)' = a*b² in a group G, then ab = ba.

A: Given,ab2=a2b2To prove: ab=ba

Q: Assume that G is a group such that for all x E G, * x = e. Prove that G is an abelian group.

A: Here we have to prove that G is an abelian group.

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Q: (a). If G is an abelian group, then (b). If G an abelian group, and group. (c). If o(a) = n, then…

A: This is a problem of Group Theory.

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: Let G = (1,-1,i,-1} Prove G is a cyclic group under the multiplication operation.

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Q: 6. Prove that if G is a group of order 231 and H€ Syl₁1(G), then H≤ Z(G). n Core

A: Given that,  G is group of order 231 and H∈syl11G. We first claim that there is a unique Sylow…

Q: Prove that a group G is abelian if and only if (ab)-1 = a-lb-l Va,bEG Attach Filo

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Q: Prove that a group G is abelian if and only if (ab) = a¬!b-1 for all a and b in G.

A: A group G is abelian if it is commutative under the operation *. In other words, G,* is an abelian…

Q: Q 12 Every subgroup of an abelian group is abelian.

A: As per our guidelines we are allowed to do 1 question at a time. Please post other questions next…

Q: prove that a group G of order p^2, where p is a prime, is abelian.

A: Suppose, G is a group of order p2 where p is prime.

Q: '. Assume that G is a group such that for all x E G, x * x = e. Prove that G is an abelian group.

A: Consider any two elements a and b in G. So, a,b,ab,ba∈G. Note that I am directly writing the…

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: Assume that G is a non-abelian simple group and that |G| < 168. Prove that G = A5.

A: Given: G is a non-abelian simple group and order of G&lt;168.  We have to prove that G≅A5 We will…

Q: Suppose G is a group in which all nonidentity elements have order 2. Prove that G is abelian.

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Q: Prove that a group is abelian if and only if (ab)-l = a-lb-1 va,bEG Attach File Browse Local Files

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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 (G1, •) and (G2 , *) be two groups and p : G1→ G2 be an isomorphism. Then * G2 might not be…

A: G2 is abelian if G1 is abelian.

Q: 1. AUT(G) := {p: G → G|p is an automorphism} Prove: (AUT(G),º) is a group (under composition)

A: Given: Aut(G)={ φ:G→G| φ is an automorphism}

Q: Let G1 and G2 be groups. Prove that G1 × G2 is abelian if and only if G1 and G2 are both abelian.

A: Abelian groups :  A group 'G' is abelian if ,                  ab = ba   ∀ a,b ∈ G  Direct product:…