Given that f be a homomorphism of a group G onto a group G' and H= Ker(f), K' is any normal subgroup of G' and defined a set K such that K = {x E G: f(x) € K') = f-¹ (K') then we have to prove that K is a normal subgroup of G containing H and G/K = G/K'.

Q: Show that SL(n, R) is a normal subgroup of GL(n, R). Further, by apply- ing Fundamental Theorem of…

A: Suppose, ϕ:GLn,R→R\0 such that ϕA=A for all A∈GLn,R Now, sinceA∈GLn,R if and only if A≠0 Now, we see…

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Q: Find the right cosets of the subgroup H in G for H = ((1,1)) in Z2 × Z4.

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Q: If f is a homomorphism of a group G into a group G' with Kernal K. Let a € 6 be such that f(a) = a'…

A: We are given that f : G → G' is a homomorphism. Kernel of f is K.     => for all elements x in K,…

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Q: (5) Let G be a group. Let H be a subgroup of G and K be a normal subgroup of G. Prove that HK is a…

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Q: Let G be a group and N is a normal subgroup of G. Let f be a mapping from G to H/N defined by f(x) =…

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Q: Prove that if H is a normal subgroup of G st H and H/G are finitely so is G.

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A: Let G be a group, G' be its commutator and N be a normal subgroup of G such that G/N is abelian.…

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Q: 10. Let G be abelian and let H be a subgroup of G. Show that G/H is abelian.

A: We have to prove that given theorem.

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Q: Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

A: Given that Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

Q: Let Φ: G to H be a group homomorphism such that Φ(g) ≠ eH for some non-identity element g in G,…

A: a) Let G and H be finite groups andϕ : G → H a homomorphism. Then|ϕ(G)| · |Ker (ϕ)| = |G|  (By…

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Q: subgroup of G'. Prove that f(H) is a normal subgroup of G' and .Let f: G G' be an onto homomorphism…

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Q: Let G be a group and N is a normal subgroup of G. Let f be a mapping from G to H/N defined b f(x) =…

A: Given :  G be a group and N is a normal subgroup of G. Let f be a mapping from G to H/N defined by…

Q: Let K be a subgroup of G and let H = {g : gKg^-1} = K. Show that H is a subgroup of G

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Q: 4. If p: G → G' is a homomorphism, prove that ker ý is a normal subgroup of G.

A: To prove that kernal is a normal subgroup of G

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Q: If f is a homomorphism of a group G into a group G' with Kernal K. Let a € G be such that f (a) = a'…

A: We are given that f : G → G' is a homomorphism.  Kernel of f is K.      => for all elements x in…

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A: We have to prove given result:

Q: 20. Suppose that G and G' are abelian groups such that G = H₁ H₂ and G' = H₁ H₂. If H₁ is isomorphic…

A: Step 1: Step 2:Step 3:Step 4:Step 5: