Group Work 2 1. Let G be an abelian group under multiplication with the identity e. Show that H = {x² | x E G} is a subgroup of G. (use one-step subgroup test) %3D 2. Show that Z(G) is a subgroup of G. (Use two-step subgroup test)

Q: . Let p: G → H be a group homomorphism. a) p(1g) = 1µ; b) p(a-1) = [9(a)]-1,va E G;

A: We have: to use definition of homomorphism:

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A: To prove: (a)Every element in ℚℤ has finite order. (b) Every non identity element in ℝℚ has infinite…

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Q: 2(a) 9. Let G be a group. Show that Z(G) =

A: we know that ZG of a group G is the subset of elements of G that commute with every element of G.…

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Q: Don't use chat gpt It

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Q: Suppose G is a group of order 48, g € G, and g" = €. Prove that g = ɛ.

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Q: estion 3. Suppose G is a group with order 99. Prove that G must have an element of order 3.

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

A: Solution is given below

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Q: (3) Chapter 2, exercise 8.5: A finite group contains an element x of order 10 and also an element Y…

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Q: 2. Finite, abelian (and additive), not isomorphic to Z₂ or Z2 x Z2, and also for every element a,…

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Q: 1. Consider the group U(5). (a) What is |U(5)|? (b) For each a ∈ U(5), find |a|, , and | |   2.…

A: U(n) - Set of all positive integers less than n and relatively prime to n. Then U(n) is a group of…

Q: n 3. Suppose G is a group with order 99. Prove that G must have an element of order 3.

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Q: Aut (2) E Z..

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Q: 3. Show that Q has no subgroup isomorphic to Z2 × Z2.

A: The objective is to show that ℚ has no subgroup isomorphic to ℤ2×ℤ2