Let G be a group with no proper, nontrivial subgroups and assume that |G| > 1. Prove that G must be isomorphic to Z, for some prime p.

Q: Let G be a group and let H and K be subgroups of G so that H is not contained in K and K is not…

A: Given that G be a group and H and K are two subgroup s.t H is not contained in K and  K is not…

Q: 1. Let p be a prime. A group G (either finite or infinite) is called p-group if the order of every…

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Q: Let G be an Abelian group with identity e and let n be some fixed integer.Prove that the set of all…

A: Given: G be an Abelian group with identity e and n is a fixed integer.To prove: The set of all…

Q: (3) Let H be a subgroup of a group G. Let a E G. Prove that a¹Ha = {a¹ha he H} is a subgroup of G.

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Q: Let H be a proper subgroup of a finite group G. Show that G is notthe union of all conjugates of H.

A: Let H be a proper subgroup of a finite group G. We how prove that G is not the union of all…

Q: Let G and H be groups, and let N be a normal subgroup of G. Decide whether each of the following two…

A: (a) The statement "The group  is abelian if and only if G and H are both abelian" is…

Q: Let G be a group and suppose that G only has finitely many distinct subgroups. Show that |G| < ∞.

A: If possible  let G is of infinite  order .In particular let  G = {a1,a2,...,an,...} .Let us consider…

Q: Let H be a subgroup of a group G with the property that for all a andb in G, aHbH = abH. Prove that…

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Q: Let G= be a finite cyclic group of order n. Suppose that n = dq where d and q are non-negative…

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Q: Let G be a group of order 90. show that G has at most one subgroup of order 45

A: Given: G be a group of order 90

Q: Let : G → H be a group homomorphism. Show that is a subgroup of H. Im(6) := {p(g) | g = G}

A: Let G,∘ and H,* be two group and ϕ:G→H be a homomorphism. Let ϕG is a non-empty subset of H. Since…

Q: Suppose that G is a finite group and let H and K be subgroups of G. Prove that |HK| = |H||K|/|HN K|.

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Q: Let G be a group and let A1, A2, B1, B2 be the proper nontrivial subgroups of G. Suppose that A1×A2…

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Q: Let G be a group and SSG a subset such that st=ts for all s, t in S. Prove that the subgroup (S) < =…

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Q: Prove that any subgroup H (of a group G) that has index 2 (i.e. only 2 cosets) must be normal in G

A: To show that H is a normal subgroup we have to show that every left coset is also a right coset. We…

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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