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: 1. If (G,*) a group such that a² = e for all a € G. Prove that G is abelian. 2. Define an operation…

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Q: (ii) Let K={(), (1 2), (3 4), (5 6), (1 2)(34), (1 2) (5 6), (3 4) (5 6), (1 2)(3 4)(5 6)} with the…

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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 G be an abelian group with identity e. Prove that is a subgroup of G. H = {g E G|g" = e for some…

A: G is an abelian group and H is a subset of G.

Q: Define ∗ on Z by x ∗ y = x + y + 4. Then: (d) Show that every element of Z is invertible with…

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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: 2. Let G be a group and H1, H2 <G subgroups. (a) Suppose |H1| = 12 and |H2 = 28, prowe H1 n H2 is…

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Q: Prove that the group G = [a, b

A: Given, the group G=a, b with the defining set of relations…

Q: Let G be an abelian group and m be a  positive integer.  Then the set             mG={mn∣x∈G}  and…

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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: (3) Write down all proper subgroups in the group Z3 × Z3.

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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: 1. Let G and G' be two groups. Let p: G G' and :G G' be two homomorphisms then prove that H = (x EGI…

A: Using the definition of subgroups, we will first show that H is a subgroup of G. And then we will…

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 homomorphism f of a group G into a group G' is an ismorphism iff ker f = {e}.

A: We have to proof that a homomorphism f of a group G into a group G' is an isomorphism if and only if…

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: (a) Show that a group G is abelian, if (ab)² = a²b², for a, b € G[C.

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Q: 1. Prove or disprove (a). The order of the element (2, 3) € Z6X Z15 is 5. (b). The group Z7X Z17 x…

A: As per the guidelines, I am supposed to do the first three sub parts. Kindly post the other…

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: A homomorphism f of a group G into a group  G' is an ismorphism iff ker f={e}.

A: We have to proof that a homomorphism f of a group G into a group G' is an isomorphism if and only if…

Q: (c) Let G, H be groups (i) Ifo: G H is a homomorphism, then prove that o(2") = (o(x))" for all ne Z+

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Q: Let G be a non-abelian simple group. (i) Write the action of G on itself via conjugation. (ii) For…

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Q: (a) G = Z where a * b = a + b – 1 for all a, b EG (b) G = {r € Q |r > 0 } where a * b = ab for all…

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Q: Month Temperature 1 W|N 2 3 4 5 6 7 8 9 10 11 12 14 13.1 18 20 26 27.9 30 26.9 24 21 16 13.1 20 35…

A: The given data, Month  Temperature 1 2 3 4 5 6 7 8 9 10 11 12 14 13.1 18 20 26 27.9 30 26.9…

Q: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,

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: If (G, ) is a group with a = a for all a in G then G is %D abelian

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Q: 4. Prove that if G and G' are isomorphic groups, then Aut(G) and Aut (G') are also isomorphic.

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Q: Prove: If Aut(G) is cyclic, then so is any subgroup of it, in particular Inn(G). Inn(G) G/Z(G) where…

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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: If (G, * ) is a group with a a for all a in G then G is abelian

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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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