Give an example f a homomorphism a) Give an example of f Such hat /Kerf/= 6 a homomaphism 5) Give an exumple f a non- abelian group o order more than 8 Which meps a ndmal subgraup n to sub yroup

Q: Exercise 3 (Second isomorphism theorem). Let G be a group, N < G, H < G. Prove that (а) НnN4Н. ( b)…

A: Let, G be group N∆G, H&lt;G Part(a): H∩N∆H⇒e∈H and e∈H∩N⇒e∈H∩N Let, g∈H∩N then for every x∈H…

Q: If in be any given integery, show that the mapping of: Com to defined by f(z)=z² is on endomorphism…

A: If n be any given positive integer, we have to show that the mapping  defined by…

Q: Additive R group and multiplicative R= {aER:ax} of the group is isomorphic D:RsR, $6x) =2* Show with…

A:

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: give an example of a finite, non-cyclic abelian group containing a container of order 5

A: Take the abelian group G=Z5×Z5 of order 25 whose every element (except identity) is of order 5 and…

Q: xx+24=5x² fpyonbouy of anyos

A: We have xx+24≤5x2 Consider:…

Q: Find x and y for which the following equation is satisfied: x+2 = - 5+i i-5

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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: 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: 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: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a = e a^4 =…

A:

Q: An endomorphism of the group G is a homomorphism : G → G. Let End (A) be the set of all…

A: Using the definition of ring set of endomorphism from A is form a Ring.

Q: (1) State the structure theorem for finite generated Abelian group. Using this theorem to classify…

A: To find the all finite generated Abelian group of order 120: Structure theorem for finite generated…

Q: G group is called meta- Abelian if it has an Abelian subgroup N which is normal in G G and is…

A: (Solving the first question as per our guidelines) Question: Show that S3 is meta-Abelian.

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: G group is called meta- Abelian if it has an Abelian subgroup N which is normal in G and is Abelian.…

A: Given that We say a group G to be meta- Abelian if it contains an abelian subgroup denoted by N…

Q: prove: let g be a group, if g is abelian then (ab)^2 = (a^2)(b^2)

A: Given g is an abelian group.

Q: Show that every group of order < 60 is solvable.

A:

Q: (a) List all the finite abelian groups (upto isomorphism) of order 1800.

A:

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

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Q: Suppose that f: G → G such that f(x) and only if = axa. Then f is a group homomorphism if a = e a^4…

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Q: iv Sketch the Caley Graph of the additive Group of direct product Z3× Z4 with respect to the…

A: Consider the conditions given in the question. Clearly from the hint a torus is involved in the…

Q: Find the smallest positive integer n such that the number of isomorphism classes of Abelian groups…

A: We know any group of prime order is cyclic.  Hence, any group of order 1, 2, 3, 5, 7, 11, or 13 is…

Q: Suppose that f:G - G such that f(x) = axa. Then fis a group homomorphism if and only if a^4 = e O…

A: (1) It is given that the function is fx=axa2. For homomorphism, the function follows the condition…

Q: Lemma : The set of all automorphism of K from a group under composition of mappings.

A:

Q: Suppose that f: G G such that f(x) and only if = axa. Then f is a group homomorphism if -> a = e a^3…

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Q: the product of the elements is the identity. 7 has order 2. of order 2.

A: Let us prove above result

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,…