THEOREM: (Cayley) Let (G, *) be a group and a E G. Then 1. The map Aa: G G defined by A(x) = a*r is a permutation in SG, where SG = {all bijections f: G G}. 2. H = {Aa: a € G}< Sc- 3. G H.

Q: 5. E Prove that G is an abelian group if and only if the map given by f:G G, f(g) = g² is a…

A: The solution is given as

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:

Q: 1. Let G be a group and let a € G. Define the centralizer of a in G, C(a), to be the set of all…

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Q: Let G = &lt;x, y | x2n = e, xn = y2, y-1xy = x-1&gt;. 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…

Q: If R be the additive group of real numbers and R+ the multiplicative group of positive real numbers…

A: Homomorphism:Let (G,∗G) and (H,∗H) be two groups.A mapping f:G→H is said to be a homomorphism if it…

Q: 15. Let G be a group and let g be in G. Define a function :G → G by ,(x) = gxg Show that :G→G is…

A: Let G be a group and let g be in G . Define a function ϕg : G→G by ϕgx=gxg-1. To prove: ϕg : G→G is…

Q: 4. Let G be a group and g e G. Prove that the function f: G G given by f(x) = gx is a bijection.

A: The solution for the asked part , is given as

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: 6. Let : (G,*) → (G',') be a group isomorphism, and let a € G. Prove that (a). G is abelian if and…

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

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Q: R\{-1} by a * b = a + b+ ab. Show that (R \ {-1},*) is an 3. Define the operation abelian group. *…

A: The solution for the asked part , is given as

Q: Show that D^2 = {(x, y) ∈ E^2: x^2+y^2 ≤ 1} ⊂ E^2 and the space containing a single point are…

A: In algebraic topology, two topological spaces are said to be homotopy equivalent if there exists a…

Q: Question 1 Consider the groups Z² = Z× Z and Z³ = ZxZxZ and the map ƒ : Z³ → Z² defined by the rule…

A: We know that Given groups G (with operation *) and H (with operation #), the direct product G × H is…

Q: Dicn = (r,s | 2,¹s-2, rsrs-1). (a) Prove that Dic, has at most 4n elements. (In fact, it has exactly…

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Q: Let y: 72 → 14Z be the mapping given by the rule o (a) = 2x for any a e 7Z. a. Show that o is a…

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Q: 51. Let N be a normal subgroup of G and let H be a subgroup of G. If N is a subgroup of H, prove…

A: According to our guidelines we can answer only first question and rest can be reposted. Not more…

Q: SIZ): Set of all fiZ-→Z which are both one-one S(Z): on ti and S(Z) is a group under composition of…

A: Given, S(ℤ)=f|f:ℤ→ℤ, 1-1, onto is a group under composition of function. And H=fn:ℤ→ℤ|fn(x)=n+x.…

Q: (c) If K is a subgroup of G, then p(K) is a subgroup of H. Given: K < G (d) If K' is a subgroup of…

A: I have mentioned the test used to prove a subset to be a subgroup, as a note above. You can skip it…

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: Given the graph of the polynomial function, determine its equation: 6- 4 2- 0 -2- -4- H 2 3 4 X

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Q: Are the groups (co ટીડ, defined in class, as answer.

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Q: True or False. If true, explain. If false, give a counterexample. (f) Let ϕ : G → G' be a group…

A: According to the given information, it is required to tell whether the given statement is true of…

Q: 3. Let f : (R\{0},-) (R\{0}, -) be group homomorphism defined by f(a) = |a|. Then ker(f) %3D =...

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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: 9*. Use Lagrange's theorem to prove: If g E G, G a finite group, then gG| = 1.

A: To prove the given statement, assume the following.

Q: (3) For any group G, Z(G) ≤ [G, G].

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Q: (a) Consider the symmetric group Sx on the set X = R-{0, 1}. Let f be the function defined by the…

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Q: Assume that G is a group such that for all x E G, * x = e. Prove that G is an abelian group.

A: Here we have to prove that G is an abelian group.

Q: Let C be a group with |C| = 44. Prove that Cmust contain an element of order 2.

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Q: Let(G,*) and (H,#) be a groups if f: G H and g: H G are homomorphism such that gof = IG.f og = IH…

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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: 1. Let A be an abelian group, and let G = A x A (Cartesian product), which also is a group. Let 6:…

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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: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a = e O a^4…

A: Given that f:G→G be a function such that fx=axa

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

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Q: Let R* denotes the set of positive real numbers and let f: R+ R+ be the bijection defined by ƒ (x) =…

A: In this solution, we will examine the properties of the binary operation ⊗ on the set of positive…