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: Suppose that G is a finite abelian group.  Prove that G has order pn where p is prime, if and only…

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Q: Exercise 3.5.2 Suppose that G, G' are finite groups and T:G→G' is a group homomor- phism. Show that…

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Q: Let m, n ɛ Z+ such that gcd(m, n) = 1. Suppose there are M and N abelian groups, up to isomorphism,…

A: Given that m,n∈ℤ+ such that gcdm,n=1. Suppose that there are M and N abelian groups, up to…

Q: Exercise 7.2.10. Let G be a group of order pq for primes p and q. Prove all proper subgroups of G…

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Q: ii) Does there exist a group G such that G/[G,G] is non-abelian? Give an example, or prove

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Q: Let m, n e Zt such that gcd(m, n) = 1. Suppose there are M and N abelian groups, up to isomorphism,…

A: We have M and N are abelian groups of order m and n. so, we know that MN is abelian group.

Q: IL Show thai al subgroup of erder 125. 18. Show that there are no simple groups of order 255 =…

A: By using syllow theorems the solution is given as follows :

Q: abelian group is not cyclic if and only if it contains a subgroup isomorphic to Zp×Zp

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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: Give an example, with justification, of an abelian group of rank 7 and with torsion group being…

A: consider the equation

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: prove that a group of order 45 is abelian.

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Q: 1. Show that the automorphism groups of two isomorphic groups are isomorphic. Let G be a group and g…

A: If ϕis an isomorphism from a group G onto itself, then it is referred to as automorphism.…

Q: Exercises 3(a) Prove that the groups $ and D6 are isomophic ; (e) Preve that the additive and the…

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Q: Prove that any two groups of order 3 are isomorphic.

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Q: Suppose that G is a finite Abelian group. Prove that G has order pn,where p is prime, if and only if…

A: We have given G is a finite Abelian group, then we have to prove that G has order pn,where p is…

Q: 3. Let G, H, K be finitely generated Abelian groups. (i) Show that if G G HH then G= H. (ii) Show…

A: We will solve this by the help of isomorphism

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: 4. List all of the abelian groups of order 24 (up to isomorphism).

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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: Show that the units in Commutative ring unit element forms with an abelian group wv.t multipli…

A: let the two units of ring  R are u  and v let u-1 and v-1 also belong to R. therefore, uu-1=1 and…

Q: b. Find all abelian groups, up to isomorphism, of order 720.

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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.

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Q: 5. Prove that no group of order 96 is simple. 6. Prove that no group of order 160 is simple. 7. Show…

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Q: (1) Z/12Z

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Q: Give three examples of groups of order 120, no two of which areisomophic. Explain why they are not…

A: Let the first example of groups of order 120 is, Now this group is an abelian group or cyclic group…

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: Describe Aut(Zp), the automorphism group of the cyclic group Zp where p is prime. In particular find…

A: Solve the following

Q: 4. Let G be a finite group of order 4 containing no element of order 4. Explain why every…

A: See below

Q: By applying Fundamental Theorem of group homomorphism, show that the quotient groups GL(n, R)/SL(n,…

A: It is given that,  GL(n, ℝ) is the set of general linear group of all n×n matrices and  SL(n, ℝ) is…

Q: Prove that the 2nd smallest non-abelian simple group is of order 168.

A: Introduction- An abelian group, also known as a commutative group, is a group in abstract algebra…

Q: A group extension A , G , B of a group A by a group B consists of a group G, an injective group…

A: Given Groups A and B and functions :f:B×B→A, ϕ:B→A at A Normalization Conditions:  ϕ(1)=1…

Q: 3) Let G be a group. Show that if Aut (G) is cyclic, then G is abelian.

A: Solution is given below

Q: 3.10. Let G be a group containing at most three elements. Show that G is abelian.

A: Here given that G is a group contains at most three that is G contains 1 or 2 or 3 elements. Case -…

Q: (G, .> be a group such that a.a = e for all a E G. Show that G is an abelian grou 2. Let

A: We have to solve given problem:

Q: 5. Let (G, *) be a cyclic group, namely G=(a). Prove that (a). if G is finite of order n, then G…

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Q: Let G be an Abelian group and let H be the subgroup consisting ofall elements of G that have finite…

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Q: Let G be a finite group of order 4 containing no element of order 4. Explain why every nonidentity…

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Q: 3. Let Dg be the group of symmetries of a regular octagon. No element of Dg of order 2 is an element…

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Q: 9. Show that the two groups (R',+) and (R'- {0}, -) are not isomorphic. | 10. Prove that all finite…

A: Two groups G and G' are isomorphic i.e., G≃G′, if there exists an isomorphism from G to G'. In…

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: 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 Of…

A: Solution:Given    G be a group∀a,b,c,d and x in G

Q: (iv) Does there exist a group G such that [G, G] is non-abelian? Give an example, or prove that such…

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Q: Let M be a group (not necesarily an Abelian group) of order 387. Prove that M must have an element…

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Q: 1) Prove that if G is an abelian simple group then G = Z„ for some prime p. (Do not assume G is…

A: By assuming G is simple abelian group we have to prove that G is isomorphic to Zp.

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

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Q: 2 Suppase G is an abeli Group of Order 6 Contauining an element' of order 3. Prove G is Cyclic. 2…

A: 1)  Let , G  be an abelian group of order 6. Consider , a ∈ G  of order 3. By definition order of an…

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.