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

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

A:

Q: Exercise 3.2.6 Show that if G and H are isomorphic groups, then G commutative implies H is…

A: A group G is called Commutative if for any a,b in G imply ab=ba

Q: All groups of order three are isomorphic.

A: All groups of order three are isomorphic.

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

A:

Q: Let G be a group and g E G. Show that Ø:Z→G by Ø(n) = g" is a homomorphism and isomorphism.

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

A: G2 might not be abelian even if G1 is abelian.

Q: Suppose T: G-> H is group isomorphism and S:H-> K is also a group isomorphism. SHow the…

A:

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

A:

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: If a is an element of order 8 of a group G, and

A: Let G be a group. Let a is an element of order 8 of group G. That is, a8=e          where e is an…

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: b. Find all abelian groups, up to isomorphism, of order 360.

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

A: A one-to-one correspondence φ : G1 → G2 between groups G1 and G2 is called a groupisomorphism if…

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…

Q: Let the order of group G =8, show that G must have an element of order 2.

A: Let G be a group and O(G)=8 Also Let a∈G be an arbitrary element other than identity. Then the…

Q: Let G be a finite group. Then G is a p-group if and only if |G| is a power of p. We leouo the

A: Given G is finite group and we have to prove G is a p-Group of and only if |G| is a power of p.

Q: Prove that the additive group L is isomorphic to the multiplicative group of nonzero elements in

A:

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

A:

Q: Suppose G is a group of order 48, g € G, and g" = €. Prove that g = ɛ.

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Q: suitte (b) Given two groups (G,) and (H, *). Suppose that is a homomorphism of G onto H. For BH and…

A:

Q: If a is an element of order 8 of a group G,

A: Let G be a group. Let a be an element of order 8 of group G. That is, a8=e  where e is an identity…

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: Let(G,*) and (H,#) be a groups if f: G H and g: H G are homomorphism such that gof = IG.f og = IH…

A:

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

A: Consider the provided question, Let G be a group with the order of G = pq, where p and q are prime.…

Q: Let (G1, •) and (G2 , *) be two groups and p : G1 – G2 be an isomorphism. Then * O G2 is finite if…

A: We will use basic results of group theory

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

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Q: Show that the multiplicative group Zfi is isomorphic to the additive group Z10.

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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 let Z(G) be the center of G. Then the factor group G/Z(G) is isomorphic to the…

A: We have to check G/Z(G) is isomorphic to group of all inner automorphism of G or not. Where, Z(G) is…

Q: For any group G, GIZ(G) is isomorphic to Inn(G)

A:

Q: 1 c = |G| gEG

A: The centralizer of a group G is the set of all elements in G that commutes with a particular element…

Q: Prove that if G is a finite group and a ∈ G, then the order of a divides the order of G.

A: Given that G is a finite group and a∈G. To prove the order of a divides the order of G: Let the…

Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: Let G be a group and let A1, A2, B1, B2 be the proper nontrivial subgroups of G. Suppose that A1×A2…

A:

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: Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = {g E G| $(g)…

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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…

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Q: If H is the subgroup of group G where G is the additive group of integers and H = {6x | x is the…

A: Let H is a subgroup of order 6 . Take H=6Z where Z is integers.

Q: A group homomorphism f:G G'is called Epimorphisum if f is 1-1 False True

A: We have to tell A group homomorphism f : G → G' is called Epimorphism if f is 1-1  Is true or false…

Q: a) Is there any relation between the automorphism of the group and group of permutations? If exists,…

A: An automorphism of a group is the permutation of the group which preserves the property  ϕgh=ϕgϕh…

Q: suppose H is cyclic group. The order of H is prime. Prove that the group of automorphism of H is…

A:

Q: (a) If G is abelian and A and B are subgroups of G, prove that AB is a subgroup of G. (b) Give an…

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Q: Suppose that ¢ : G → G' is a group homomorphism and there is a group homomorphism : G' → G such that…

A:

Q: 1,) and (G2,*) be two groups and →G2 be an isomorphism. Then *

A: given that G1,. and G2,*are two groups  and φ:G1→G2  be an isomorphism

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: 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.