Let G be any group and A (G) the set of all 1-1 mappings of G, as a set, onto itself. Define La: G→ G by La(x) = xa¹. Prove that: (a) La E A (G). (b) LaLb= Lab. (c) The mapping : GA (G) defined by (a) = La is a monomor- phism of G into A (G).

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Q: 7) Define a map ø1 from Z3 x Z → Z4 as follows: 01(m, n) = 4(m + n). Is ø1 a well-defined map? (If…

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Q: Let G be a group and g E G. Show that Ø:Z→G by Ø(n) = g" is a homomorphism and isomorphism.

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Q: find the fundamental group of X := {(x, y, z) = R³|(x² + y²) (y² + z²)(x² + z² − 1) = 0}

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Q: 6. Use the Fundamental Homomorphism Theorem to prove that the following groups are isomorphic. (a)…

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A: Center of the given group is Zn itself.Explanation:Step 1: Step 2: Step 3: Step 4:

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Q: Let G be a group and let H and K be normal subgroups such that HnK={e}. Let : G→G/HxG/K be the map…

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Q: 1*. Let f G H be a group homomorphism. Prove: (a) If l is the identity of G then /(lg) is the…

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

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Q: Exercise2: Let T = {(a, b) | a, bE R\{0}} and * defined on T E %3D (a, b) * (c, d) = (ac, bd) for…

A: Yes it is group. Let  x1=a1,b1 , x2=a2,b2 and x3=a3,b3belong to T. Then, x1*x2=a1,b1*a2,b2=a1a2,b1b2…

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Q: 3) Let G and H be two groups with e and e' as identities. Define P = G x H by P = G x H = { (g,h) |…

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Q: Let G be a group. a) Define the center Z(G) b) Show that Z(G) is a subgroup of G c) Show that Z(G)…

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Q: Exercise 3.1. Let G be a group. Given g € G, define ig : G → G by ig(x)=gxg, Vx Є G. (1) Show that…

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Q: Exercise 4.1. (1) Let : G → G' be an isomorphism between two groups G and G'. Suppose G is cyclic.…

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