Let G be a group with the order of G = pq, where p and q are prime. Prove that every proper subgroup of G is cyclic.

Q: Let G be a group, let H < G, and let x E G. We use the notation xHx1 to denote the set of elements…

A: In a group G, two element g and h are called conjugate when h = x g x-1 for some x ∈ G For an…

Q: Let (G,*) be a group of order p, q, where p, q are primes and p < q. Prove that (a). G has only one…

A: It is given that G, * is a group of order p·q where p, q are primes and p&lt;q. Show that G has only…

Q: Let G be the subgroup of GL3(Z₂) defined by the set 100 a 10 bc1 such that a, b, c Z₂. Show that G…

A: The given set of matrix is 100a10bc1 where a, b, c∈ℤ2. To find: the group to which the given set is…

Q: Q2: Let (G,) be a commutative group, and let the set H consist of all elements of G with finite…

A: Given a group G and a set H of G with the given conditions. We need to show that H is a normal…

Q: If H is a subgroup of a group G such that (aH)(Hb) for any a, bEG is either a left or a right coset…

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Q: Let G be a group of order 24. Show that G is solvable

A: Let G be a group of order 24. Show that G is solvable

Q: If Φ is a homomorphism from Z30 onto a group of order 5, determinethe kernel of Φ.

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Q: let G be a group and H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

Q: If H is a Sylow p-subgroup of a group, prove that N(N(H)) = N(H).

A: Let G be a finite group and H be the subset of G. Then, normalizer of H in G, when we conjugate H…

Q: Suppose that G is a finite group with the property that every nonidentityelement has prime order…

A: This can be proved by lemma that states If G is abelian with the property that every nonidentity…

Q: Prove that if H is cyclic, then ø[H] is also cyclic.

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Q: If H is a normal subgroup of a finite group G and |H| = pk for someprime p, show that H is contained…

A: H is a normal subgroup of a finite group G and |H| = pk for some prime p.

Q: Let G be the subgroup of GL3(Z2) defined by the set 100 a 10 b C 1 that a, b, c Z₂. Show that G is…

A: Given: G is the subgroup of GL3ℤ2 which is defined by the set of matrix 100a10bc1 where a, b, c∈ℤ2 .…

Q: Let G be a group with IG|=187 then every proper subgroup of G is: * O Non abelian O cyclic O None of…

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

A: Given G be a finite group. Let x∈G. Define: xG=gxg-1: g∈G Let G:G=m. Define a map π:G→GG:G defined…

Q: Let G be a group with IG|=187 then every proper subgroup of G is:* O Cyclic None of these Non cyclic…

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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…

A: Given orders of subgroup 10 18 30

Q: Let H be the set of all elements of the abelian group G that have finite order. Prove that H is a…

A: Let H be the set of all elements of the abelian group G that have finite order.  Prove that H is a…

Q: Let G be a group of order 24. If H is a subgroup of G, what are all the possible orders of H?

A: Given, o(G)=24 wherre H is a subgroup of G from lagrange's theoram: for any finite order group of G…

Q: Prove that any two Sylow p-subgroups of a group G are conjugate.

A: To prove that any two sylowp-subgroup of a group G are conjugate

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 be a group with IG|=209 then every proper subgroup of G is: * O Non abelian O Non cyclic O…

A: See the detailed solution below.

Q: Prove that every subgroup of Z is either the trivial group, {0}, or nZ = {nx | x E Z} for some n E…

A: To prove: That every subgroup of ℤ is either the trivial group{0} or nℤ=nxx∈ℤfor some n∈ℕ. Proof:…

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 group with IG|=221 then every proper subgroup of G is: None of these O Cyclic O Non…

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Q: Let H and K be subgroups of a group G. Prove that HNK is a subgroup of G.

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Q: Let p : G → G' be a group homomorphism. (a) If H < G, prove that 4(H) is a subgroup of G' (b) If H <…

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Q: Let G be a group. Prove that Z(G) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called the…

Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: Let G be a group, and N ⊆ Z(G) be a subgroup of the center of G, Z(G). If G/N, the quotient group is…

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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: If H is a subgroup of a group G such that (aH)(Hb) for any a, b eG is either a left or a right coset…

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Q: Let G be a group and g E G. Prove that if H is a Sylow p-group of G, then so is gHg-1

A: It is given that, G is a group and g∈G. To sow that if H is a sylow p-subgroup of G, then so is…

Q: Suppose that G is a group and |G| = pnm, where p is prime and p &gt; m. Prove that a Sylow…

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Q: Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no subgroup…

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Q: G is a finite group of order IGI =pqr with p< q <r prime

A: Solution

Q: In the group (Z, +), find (-1), the cyclic subgroup generated by -1. Let G be an abelian group, and…

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Q: Let G be a group with |G|=187 then every proper subgroup of G is: * Non abelian Non cyclic None of…

A: See the detailed solution below.

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: Prove that every group of order 78 has a normal subgroup of order 39.

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Q: If H₁ and H₂ be two subgroups of group (G,*), and if H₂ is normal in (G,*) then H₂H₂ is normal in…

A: When a non-empty subset of a group follows all the group axioms under the same binary operation, the…

Q: Let G be a group with |G|=221 then every proper subgroup of G is: ONone of these Non abelian Non…

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Q: Let a be an element of a group G such that Ord(a) = 30. If H is a normal subgroup of G, then Ord(aH)…

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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 G be a group with no proper, nontrivial subgroups and assume that |G| > 1. Prove that G must be…

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Q: 6. Let G be a group of order p², where p is a prime. Show that G must have a subgroup of order p.

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Q: Let G be a group of order 24. Suppose that G has precisely one subgroup of order 3, and one subgroup…

A: Theorem : If a group G is the internal direct product of subgroups H and K, then G is isomorphic to…