Let G be a group and a be an element of this group then necessarily: * Jal<|G| O lal2|G| O lal=|G|

Q: Let G be a group and let a,b element of G such that (a^3)b = ba. If |a| = 4 and |b| = 2, what is…

A: see below the answer

Q: 3. Let G be a group and let SG be the set of all permutations on G. For each g € G, define Kg: GG as…

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Q: Give an example: The product of two solvable groups need not to be solvable?

A: it is clear that s2 is solvable because it is abelian

Q: 9) Let H be a subgroup of a group G and a, be G. Then a e bH if and only if O b-la e H O ba e H O…

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Q: Let G be a finite group of order n . Let g be an element of G with order k. Prove that k|n

A: To prove ''Let G be a finite group of order n . Let g be an element of G with order k. Prove that…

Q: 3. Suppose that a group G contains elements of orders 1, 2, 3, 4, ..., 8, 9. What is the minimum…

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Q: Give an example to show that there exists a group with elements a and b such that la] = 0, \b| = ∞,…

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Q: 7. Let G be an infinite cyclic group. Give an example of a subgroup H in G such that H {e} or G. 8.…

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Q: (d) Define * on Q by a * b = ab. Determine whether the binary operation * gives a group on a given…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Let (G, *) and (H,) be groups. If group (G, *) is cyclic and & H, then (H,) is also cyclic. Select…

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Q: Let N be a normal subgroup of a finite group G.  Prove that the order of the group element gN in G/N…

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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: (5) Let (G, *) be a group with a² = e for every a € G. Show that G, a*b = b* a (that is, show that…

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Q: 10, Let (G, *) be a group and a, b, c E G. (a) Prove if a *b = a * c, then 6 = C (b) Prove if b * a…

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Q: Let G = &lt;a, b | a6 = b3 = e, b-1ab = a3&gt;. How many elements doesG have? To what familiar group…

A: Given set is G = &lt;a, b | a^6 = b^3 = e, b^-1ab = a^3&gt;.

Q: (4) Let G be a group, a, b E G such that |a| = 20, |b| = 12. Suppose (a) (b) ‡ {e}. Prove that b²a¹0…

A: &lt;a&gt;∩&lt;b&gt;≠e&lt;a&gt;∩&lt;b&gt; order either 2 or 4if (&lt;a&gt;∩&lt;b&gt;) has order 2then…

Q: Let G be a group under multiplication, G be a group under addition and o be an isomorphism from G to…

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Q: If G is a group, prove that Aut(G) and Inn(G) are groups

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Q: Is it possible to find a group operation e on a set with 0 elements? With 1 element? Explain why or…

A: The question is :: is there possible to find a group operation on a set of 0  element? Or with 1…

Q: Explain the following statement "If G is a group an a E G then o(a) = | |." 31. %3D

A: Order of element a ∈ G is the smallest positive integer n, such that an= e, where e denotes the…

Q: 2) Let G be a group and H be a subgroup of G then H x = H• y -y-. xcH. true O false

A: (a) Given that G is a group and H is a subgroup of G H.x=H.yH.x.y-1=H.yy-1H.xy-1=Hxy-1∈H Hence,…

Q: (1).* 4 and |6| = 2, Let G be a group. Let a, b EG such that |a| and a b = ba. Then determine |ba|,…

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Q: Let a be an element of a group. Suppose abs(a5) = 12. Show that abs(a) = 12 or abs(a) = 60.

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Q: 3. Let G be a finite group of order n and identity element 1. a. Show that g" = 1 for any g€ G. b.…

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Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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Q: Let G be a group and let a, be G such that la = n and 6| = m. Suppose (a) n (b) = (ea). Prove that…

A: According to the given information, let G be a group.

Q: 1)Let G be a group and H be a subgroup of G then, for any a, b e G,a = b mod H + (a + b-1) H = H.…

A: Given: To show the correct option is true or false as follows,

Q: 19. Let H and K be subgroups of a group G. Prove that gH ngK is a coset of HOK in G.

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Q: Q2 / If a, b and c are elements of a group (G,*) such that c*a=c*b then a=b.

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Q: Consider a group G with operation * and identity element e. Show that for any element a in G, there…

A: In any group G, we define a group as a set G equipped with an operation * that combines any two…

Q: Let G and H be two groups. Let a € G have order n and 6 € H have order m. Show that (a, b) € G x H…

A: G and H are two groups.a∈G have order n and b∈H have order m.

Q: 3. Prove or disprove: If G is a group, (g¬')-' = g.

A: Consider the given information: Let G is a group. To show that (g-1)-1=g