Suppose that : G H is a surjective group homomorphism. Suppose that there is an element of order 27 in H. Prove that G contains an element of order 27.

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Q: Let G be a finite group. Then G is a p-group if and only if |G| is a power of p. We leouo the

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Q: Suppose S is a nonempty subset of a group G.(a) Prove that if S is finite and closed under the…

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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 and H be groups and let ø:G → H be a homomorphism. Prove that ø(G), the image of ø, is a…

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Q: Suppose 0: G-→ G be a homomorphism of groups. Show that: (i) Kere is normal in G (ii) The mappping…

A: Normal subgroup: A subgroup N of a group G is said to be a normal subgroup of G if  gNg-1=N , for…

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

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Q: Let l be an integer greater than 1, and let G be a finite group with no element of order l. Can…

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Q: suitte (b) Given two groups (G,) and (H, *). Suppose that is a homomorphism of G onto H. For BH and…

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Q: Let G be a finite group. Let x E G and let G be the conjugacy class of r. Prove that |#C| < |[G,…

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Q: Prove that the trivial representation of a nite group G is faithful if and only if G = {1G}

A: To Prove: The trivial representation of the nite group G is faithful if and only if G =1G

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

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

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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: For each element a in the group G, define the mapping ka: G → G by ka(x) = xa-'for all x in G. %3D…

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Q: Suppose c is a conjugacy class in a с group G such that IC is finite but | c/#1. Then there exists…

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Q: Prove that if G is a finite group, then the index of Z(G) cannot be prime.

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Q: For any group G, GIZ(G) is isomorphic to Inn(G)

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Q: 1 c = |G| gEG

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Q: Let G be a group with p or more distinct elements of order p for some prime p. Show that G is not…

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Q: Prove that if G is a finite group and a ∈ G, then the order of a divides the order of G.

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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: If G is a cyclic group of order n, prove that for every element a in G,an = e.

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Q: Let G be a finite group of order 125 with the identity element e and assume that G contains an…

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Q: Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = {g E G| $(g)…

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Q: Label the following statement as either true or false. Every finite group G of order n is isomorphic…

A: True Cayley's Theorem states that every group G is isomorphic to a subgroup of the group T(S) of all…

Q: Let G be a abelian group. Prove that the set of all elements of finite order forms a subgroup of G.

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Q: Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G is cyclic.

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Q: Let G be a non-trivial group. Prove that Aut(G) × Aut(G) is Aut(G x G). a proper subgroup of

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Q: 3. Let G be a finite group with an odd number of elements. Prove that G is isomorphic to a subgroup…

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Q: Suppose that ¢ : G → G' is a group homomorphism and there is a group homomorphism : G' → G such that…

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Q: Prove that if H is normal in G and ø is onto, then ø[H] is normal in G'.

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Q: Let a and b be nonidentity elements of different orders in a group Gof order 155. Prove that the…

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Q: If G is a finite group and some element of G has order equal to the size of G, we can say that G is:…

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