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

Q: Let G be a group of order pm where p is a prime number and m is a positive integer. Show that G…

A: Let G be a group of order pm, where p is a prime number and m is a positive integer. Then,…

Q: 5. Let H be a subgroup of a group G. Prove that aH Ha-1 is a bijective mapping from the set of all…

A: Let H be a subgroup of a group G. Let f be a function from the set of all left cosets of H in G to…

Q: Let G be a finite group of order n. Let g be an element of G. Prove that gn=e.

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Q: Let G be a finite group of order n. Let g be an element of G. Prove that gn=e.

A: Given that, G be a finite group of order n. Since, g∈G and G is a group. Therefore, g2=g·g∈G.…

Q: Let r be the second largest digit in your student ID. Let G and H be groups, and let p : G → H be an…

A: Given that G and H be groups and φ : G→H be an isomorphism. Also given g∈G have order r. To show: φg…

Q: Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G. Prove that if…

A: Given that, Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G.

Q: Let H1 and H2 be finite groups, and let K = H1 × H2 be the product group. Let C1 be a conjugacy…

A: Given: H1 & H2 are finite groups. K=H1×H2 is a product group. C1 is a conjugacy class of H1…

Q: Let G be a finite cyclic group and a be an element of G of order 8 then the order of G would be

A: Would be a group.

Q: Let G be a cyclic group of order 40. The number of generators of G is

A: Explanation of the solution is given below...

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

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Q: 3. Let G be a group of order 8 that is not cyclic. Show that at = e for every a e G.

A: Concept:

Q: Let G be a group of order 42. Find all possible orders |H| for a subgroup H of G, and in each case…

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Q: Let h: G G be a group homomorphism, and gEG is an element of order 35. Then the possible order of…

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Q: 1 Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is

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Q: Let G be any group and p: G -G, (g) =g1 Vg eG. Then: (a) o is a homomorphism (b) o is a homomorphism…

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Q: Let G be a finite group of order n, positive integer, with identity e. Prove or disprove: For any g…

A: Let G be a finite group of order n, with identity e i.e, order of (G) = |G| = n .   Now let g be any…

Q: Let G be a finite group and H1, H2,…., Hk be subgroups of G. .... (a) Show that N H; = Hị n H2 n..n…

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Q: Let G be a group of order 100 that has a subgroup H of order 25.Prove that every element of G of…

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Q: 1) Let G be a finite groups whose order is not divisible by 3. Show that for every g e G, there…

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Q: Let G be a group and a ∈ G. The centralizer of  a in G is equal to the centralizer of a^-1 in G.…

A: First let us see the definition of centralizer or normalizer of a in G (definition from I.N.…

Q: If G is a finite group with |G|<120 and G has subgroups of orders 1O, 15 and 20 then the order of G…

A: Using Lagrange's theorem it can be written that if G is a finite group and B is a subgroup of G,…

Q: Let G be a finite group of order p  where p is prime. Prove that G is cyclic.

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Q: Let G be a group. The centre of G is defined as the set Z(G) = {g ∈ G : gx = xg for all x ∈ G}.…

A: Let G be a group. The center ZG of group G is defined as follows. ZG = g∈G : gx = xg , ∀x∈G We need…

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

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Q: Let G be an abelian group. Show that the set containing all elements of G with finite order is a…

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Q: 5. Prove that no group of order 96 is simple. 6. Prove that no group of order 160 is simple. 7. Show…

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Q: See attachment.

A: Given G is a group and sym(G) is the group of all permutations on G.

Q: 4. Let G be a finite group of order 4 containing no element of order 4. Explain why every…

A: See below

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

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Q: Let G be a group of finite order n. Prove that an = e for all a in G.

A: Let G be a group of finite order n with identity e.           Since G is of finite order…

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

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Q: 5. Let (G, *) be a cyclic group, namely G=(a). Prove that (a). if G is finite of order n, then G…

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Q: Let G be a finite cyclic group and a be an element of G of order 9 then the order of G would be: *…

A: Let G be a finite group and 'a' be an element  in G  s.t O(a)=9

Q: Prove that if G is a finite group and H is a proper normal subgroupof largest order, then G/H is…

A: Given: G is a finite group and H is a proper normal subgroup of largest order.

Q: Let G be a finite group and let G’ be a group. Let ø : G → G’ be a homomorphism.  Prove that | ø[G…

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Q: Let G be a group and H ≤G. Let x ∈G be an element of finite order n. Prove that if k is the least…

A: It is given that, H is a subgroup of G and |x|=n. Also  k is the least positive integer such that…

Q: If G is a finite group and a ∈ G, prove that |a| is finite. Hint: You just need to show that there…

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Q: Let G be a finite cyclic group and a be an element of G of order 8 then the order of G would be: 2…

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Q: Show that if G is a finite group with identity e and with an even number of elements, then there is…

A: Given,If G is a finite group with identity e and with an even number of elements,then there is a ≠e…

Q: 2. Let o : G → G' be a homomorphism of groups, and let H be a subgroup of G. (a) Let a E G. Prove…

A: Given   ϕ:G -&gt; G' is a homomorphism. H is a subgroup of G. The solution to the subparts is given…

Q: Let G be a finite group of order 4 containing no element of order 4. Explain why every nonidentity…

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Q: Let G be a finite cyclic group and a be an element of G of order 9 then the order of G would be: * 4…

A: Fourth option is correct.

Q: 7. Suppose G is a finite group and let n be a factor of |G|. Suppose N≤ G is the only subgroup of G…

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Q: If G is a finite group with |G|<180 andG has subgroups of orders 10, 18 and 30 then the order of G…

A: Use the fact that order of subgroup divides order of group

Q: Prove that if G is an abelian group of order n and s is an integer that divides n, then G has a…

A: G is an abelian group of order n ; And, s is an integer that divides n;

Q: Prove that the order of the group element gN in G/N divides the order of g.

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Q: 2. Let a and b be elements of a finite group G. Prove that: a. a and a¯1 have the same order. b. a…

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Q: Let G be a finite group. Show that there exist a positive integer m such that for each gE G, gm = e.

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