(1) Define the product group of (G, o) and (H, x) as the set G x H = {(g, h): gG, he H} with composition where (91, h1) (92, h₂) = (910 92, h₁ x h₂). Prove that G x H this is a group, and that it is isomorphic to H x G. What is the identity element? What is the inverse of (g, h)?

Q: Let G = <x, y | x2n = e, xn = y2, y-1xy = x-1>. Show that Z(G) ={e, xn}. Assuming that |G| =…

A: To show that Z(G) = {e, xn}, we need to find all elements in G that commute with every other element…

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A: Given a and b be elements of a group G. a = a−1 and b = b−1

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Q: 5. In each of parts (a) to (c) show that for the specified group G and subgroup A of CG (A) = A and…

A: (a) We are given that G=S3A=1,1 2 3,1 3 2 Now, we can see that 1 21 2 31 2-1=1 21 2 31 2=1 3 2 and 1…

Q: If f is a homomorphism of a group G into a group G' then (i) ƒ(e) = e', where e is the identity of G…

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Q: Let D10 = (a, b | a5 = b2 10. Assume x and y are two distinct elements of order 2 in D10. Let H =…

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Q: Let G and H be groups and consider the function f: G → G × H defined by f(g) = (g, eH ) where eH is…

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

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Q: (2) (a) Consider the set Z\ {0} of integers without zero under the operation × of integer…

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Q: Let G = : a – b = c – d, a,b, c, d E R Show that G is a group under (the usual) matrix addition.

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Q: 16. Let G be a group and let g and h be in G. Show that . - as defined in #15.

A: Given: G is a group, g,h∈G, ϕg:G→G is a function defined as ϕg(x)=gxg-1 To prove: ϕg∘ϕh=ϕgh Let x∈G…

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

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Q: Theorem 2. Let G, and G, be groups, then @ Gx G,= G, × G, (6) If H = {(a, e,)| a e G} and H, = {(e,…

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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: We have a semigroup (M, *), where M = R- {k} a a b = a + b-kab. Find also that, (M, *) is a group.…

A: Semi group will be a group if there exist an identity element and every element has inverse.

Q: Help: Let G and G' be groups with identity elements e and e' respectively and let f:G goes to G' be…

A: Let G and G' be groups with identity elements e and e' respectively and let f:G goes to G' be a…

Q: List the identity element and each other element (along with their inverses) for the group U(18)

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Q: Let (G, *), (G', *' and (G", ") be groups, and let :G G' and : G' → G" be isomorphisms. Prove that t…

A: We will use the definition of isomorphism to prove that the composition of two isomorphisms is again…

Q: (a). If G is an abelian group, then (b). If G an abelian group, and group. (c). If o(a) = n, then…

A: This is a problem of Group Theory.

Q: Let f: - be defined by f(x) = 3x-3. Prove or disprove that f is an isomorphism from the additive…

A: Consider the given information: Let f:ℝ,+→(ℝ,+) be defined by, f(x) =3x-3 To find that f is an…

Q: Let G be a group, and let xeG. How are o(x) and o(x) related? Prove your assertion

A: According to the given conditions:

Q: 2. Let G = (1, 0). Decide if G is a group with respect to the operation * defined as follows: x * Y…

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Q: Let Dn be the dihedral group of order 2n. Decide whether the following pairs of groups are…

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Q: Show that Q(√5 + √5) is Galois over Q and determine the Galois group.

A: This question investigates whether the number 5+5 is a Galois extension of the rational field ℚ, and…

Q: Hello, can you help me out with this problem? Please write a solution on a piece of paper and upload…

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Q: Hello, can you help me out with this problem? Please write a solution on a piece of paper and upload…

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Q: Let (G, *) be a group. Prove the following. (a) For each y = G, (y-¹)-¹ = y. (b) For x, y = G,…

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Q: 2. Suppose M and D are isomorphic groups. Prove that Aut(M) is isomorphic to Aut(D).

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Q: Q3: Describe the quotient group of a- (²/z, ·+) b- (2/z,+)

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Q: Let G be the set (te, ±a, ±b, ±c) where [ 1 10 Show that G form a Group under multiplication.

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Q: If G and G' are both groups then G' n G is a group.

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Q: ve Lagrange's theorem and apply it to find all subgroups of the group 7.12.

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Q: (d) Find the cosets of the quotient group (5)/(10), and determine its order.

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Q: Let F and H be subgroups of group G and let FCH. Prove (G : F) = (G : H)(H : F).

A: It is given that F and H are subgroups of G and F⊂H.

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: True or False: The group Dg has a subgroup isomorphic to D3. (a) True (b) False (c)

A: Answer :- A)False