Show that: (i) If N is a normal subgroup of an abelian group G, then G/N is also abelian. H is a group homomorphism and G is abelian, then im o is (ii) If o: G abelian.

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: (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.

Q: Let G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that…

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Q: 2. First Isomorphism Theorem. Let G be a group and H-((.s)|ge G). (a) (b) (1) G is abelian. (i) H is…

A: Given G is a group then, G×G is a group of the type such that G×G={(a,b); where a, b belong to G}…

Q: let G be an abelian Group and let H and K be a susb group of G prove that HK ={hk / h an element H…

A: Consider the provided question, Your question is missing the last statement; I think you want to…

Q: (a) of G'. Show that if y :G → G' is a group homomorphism then Im(y) is a subgroup

A: According to the given information, For part (a) it is required to show that:

Q: Let G be an abelian group, Ha subgroup of G, and n a fixed positive integer. (a) Let Hn = (aEGla"…

A: (a) Given G is an abelian group and H be a subgroup of G. For any fixed n∈ℕ, a subset of G is…

Q: 1. Define x*y over R\ {-1}by x*y = x + y +xy. Prove that this structure forms an abelian group.

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Q: (14) Let G be a finite cyclic group of order n with identity element e. Prove that if a € G, then cd…

A: The order of every subgroup of a finite group G is divisor of the order of G.

Q: (True/False) Suppose G is a group of order 25. Then every proper subgroup of G is abelian.…

A: (a) We know any group of order p2 is abelian where p is a prime number. Also, any subgroup of an…

Q: Let G and H be groups, and let N be a normal subgroup of G. Decide whether each of the following two…

A: (a) The statement "The group  is abelian if and only if G and H are both abelian" is…

Q: 3 be group homomorphisms. Prove th. = ker(ø) C ker(ø o $).

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Q: Subgroup of G. (b) Let G be an abelian group. Let T = {ge G: g = e for some positive integer n}.…

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Q: Let (G,*) be a group such that x² = e for all x E G. Show that (G,*) is abelian. (Here x² means x *…

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Q: (True/False) Suppose G is a group. Then : G x G →G, (x,y) → ay is a group homomorphism. (True/False)…

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Q: Let G be a group with IG|=209 then every proper subgroup of G is: * O Non abelian O Non cyclic O…

A: See the detailed solution below.

Q: 5. Let G be a group. Let H₁ CH₂ C H 3 C · of subgroups of G, and let H∞ = Un=1 Hn. be an infinitely…

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Q: Give an example of two non-homomorphic finite abelian groups that have the same number of elements…

A: We need to find an example of two non holomorphic finite abelian groups which have the same number…

Q: (4) Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that…

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Q: (a) Prove that a*b=a+b-a-b is an operation on the set S = R\{1}. (Show that, if a*b = 1, then one of…

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Q: Let p : G → G' be a group homomorphism. (a) If H < G, prove that 4(H) is a subgroup of G' (b) If H <…

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