a) Is there any relation between the automorphism of the group and group of permutations? If exists, give example.

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A: A group is a set of elements with a specific operation that satisfies four conditions: closure,…

Q: When do you use a t-test for independent groups and what do the numerator and deniminator in the…

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Q: Give examples of four groups of order 12, no two of which are isomorphic.Give reasons why no two are…

A: We have to give 4 examples of order 12, such that all are non-isomorphic to each other.

Q: subgroup and homomorphic image of a solvable group is solvable.

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Q: Show that isomorphism between groups is an equivalence relation.

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Q: Show that the group of automorphism of a cyclic group of degree four is of order two.

A: Claim :  The group of automorphism of a cyclic group of degree four is of order two.

Q: Let R" = {(a,, a, . .. , a) l a, E R}. Show that the mapping o: (a,, Az, ... , a) → (-q,, -a, · .…

A: Given: ℝn=a1.....,anai∈ℝWe need to show that the mapping ϕa1,.....,an→-a1,.....,-an is an…

Q: Give an example of a non-Abelian group that has exactly threeelementsof finite order.

A: Let G=Z3⊕Z be an additive group where e=0,0. Let x∈G then it can be observed that x=a,b where a∈Z3,…

Q: True or False with proof "Any free abelian group is a free group."

A: Given statement is "Any free abelian group is a free group."

Q: Prove or give counter example Every characteristic subgroup is fully invariant

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Q: Show that the order of the group of  automorphism of a finite group of  order n is a divisor of n !.

A: we have to show that the order of the group of automorphism of a finite group of order n is a…

Q: Every homeenesphic image of Arenosplic to some quodient group et

A: Statement : Every homomorphic image of a group G is isomorphic to some quotient group of G'.Our aim…

Q: Show that conjugacy is an equivalence relation on a group.

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Q: prove that Every group oforder 4

A: Give statement is Every group of order 4 is cyclic.

Q: Show that the order of the group of automorphism of a finite group of order n is a divisor of n!.

A: we have to show that the order of the group of automorphism of a finite group of order n is a…

Q: What is a quotient group and conjugacy class

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Q: et f be an isomorphism from a grou onto a Group H. et a be an element in G and the orde Fa is 5. it…

A: We have to choose the correct answer from the given option according to the given condition

Q: write why, in order to describe the homomorphic images of a group, it suffices to describe the…

A: The homomorphic images of a group refer to the set of all possible images that result from applying…

Q: Show that (up to isomorphism) A5 is the only simple group of order60.

A: Given : G=60 To Prove : G≃A5

Q: Please explain an infinite p-group, and give an example

A: Infinite p-group: Infinite p-group is an infinite group in which the order of every element is a…

Q: Let G be a group (not ncesssarily an Abelian group) of order 425. Prove that G must have an element…

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Q: Give an example, with justification, of a semi-group which is not a group.

A: A semigroup defined on a set is a binary operator which satisfies the association property of…

Q: Suppose A, B C U, (A, +, –( ),0) e Grp (A), and (B,*,-( ),0) e Grp(B). If we define (u – v) to be (u…

A: Consider the given expression: The objective is to show:

Q: If f:G\rightarrow{}G' is an isomorphism of groups, show   that the order of an element a∈G is equal…

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Q: Q4)Prove or disprove the composition of two Lie group homomorphism is a Lie group homomorphism what…

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Q: Write the Caley table of all abelian groups of order 8 Please be legible.

A: Introduction: A group is a set of elements closed under an associative operation that includes an…

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

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Q: Show that any group of order less than 60 is cyclic

A: This result is not correct. There is a group of order less than 60 which is not cyclic.

Q: Show that isomophism between group is an equivalence relation. Briefly explain and show all the…

A: Recall the definition of an equivalence relation. It satisfies reflexive, symmetric and transitive.…

Q: Consider a code whose generating matrix is 1 0 0 1 1 0 1 1 0 1 Determine the automorphism group of…

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Q: What are all abelian groups of order 81 up to isomorphism? Are there any Theorems I can use to…

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Q: Prove that there are exactly five groups with eight elements, up to isomorphism.

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Q: Show that for an abelian group, the only inner automorphism is the identity mapping, whereas for…

A: We have to show that for an abelian group, the only inner automorphism is the identity mapping,…

Q: Construct the group Z4 building an isomorphism. Z8/ and determine a group that it is isomorphic to…

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