If H is a normal subgroup of a finite group G and |H| = pk for someprime p, show that H is contained in every Sylow p-subgroupof G.

Q: 3. Prove that a subset H of a finite group G is a subgroup of G if and only if a. His nonempty, and…

A: We have to prove given property:

Q: ) Let G be a finite group , IGI=ps. p prime Prove that G cannot have two distinct and sep. subgroups…

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Q: Prove that, if H is a subgroup of a cyclic group G, then the quotient group G/H is also cyclic.

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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 K is a normal subgroup of a finite group G and S is a Sylow p-suby

A: Given that if K is a normal subgroup of a finite group  G and S is a Sylow p-subgroup of G. then K∩S…

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: If H is a cyclic subgroup of a group G then G is necessarily cyclic * True False

A: let H be a cyclic subgroup of a group G.

Q: A simple group is called G if G has no ordinary subgroup other than itself, and suppose f: G → H is…

A: The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. If we…

Q: 5. Let p and q be two prime numbers, and let G be a group of order pq. Show that every proper…

A: We have to prove that: Every proper subgroup of G is cyclic. Where order of G is pq and p , q are…

Q: (a) Prove that if K is a subgroup of G and L is a subgroup of H, then K x L is a subgroup of G x H.

A: The detailed solution of (a) is as follows below:

Q: Let H and K be finite subgroups of a group G and a E G. Then prove that |HaK| = |H||K| /|HnaKa-|.

A: Given that H and K are the finite subgroups of a group G and also an element a such that a∈G Here,…

Q: Suppose that G is a group and |G| = pnm, where p is prime andp >m. Prove that a Sylow p-subgroup…

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Q: Show that if H and K are subgroups of a group G, then their intersection H ∩ K is also a subgroup of…

A: Subgroup Test           A subset H C G of the group G will be a subgroup if it satisfies the…

Q: Let G be a group with |G| = pq, where p and q are prime. Prove that every proper subgroup of G is…

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

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Q: 9. Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no…

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Q: Suppose that K is a proper subgroup of H and H is proper subgroup of G. If |K| = 42 and |G| = 420,…

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Q: Let a be an element of a group G such that Ord(a) = 32. If H is a normal subgroup of G, then Ord(aH)…

A: As you asked multiple questions , I answered only first question.   Here we can say from a corollary…

Q: Let G be a finite group and let primes p and q ≠ p divide |G|. Prove that if G has precisely one…

A: Given that G be a finite group. Consider G be a finite group and consider prime p and q such that…

Q: Let G be a group with |G|=187 then every proper subgroup of G is:

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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…

A: (a)Suppose S is a non-empty subset of a group G. then we have to prove that if S is finite and…

Q: If N is a normal subgroup of a group G, and if every member of N and G/N have a finite order, prove…

A: Given: If N is a normal subgroup of a group G, and if every member of N and GN have a finite order…

Q: Let K be a Sylow p-subgroup of a finite group G. Prove that if x ∈ N(K) and the order of x is a…

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Q: Let |G|=pq, where p and q are prime.  If G has only one subgroup of order p and only one of order q,…

A: Given |G|=pq and G has only one subgroup of order p and only one of order q. To prove that G is…

Q: Every subgroup of a group G is normal * False True

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Q: Let H be a subgroup of G and let K=⋂φ∈Aut(G)φ(H). Show that K is characteristic in G

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Q: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A

A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.

Q: Let n be a positive integer. Show that A, is a normal subgroup of S, by choosing an appropriate…

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Q: 9. Let (G,*) be a finite group of order pq, where p and q are prime numbers. Prove that any non…

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Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * O True False

A: this is false because this is need not be true because Z4×Z6 Is not cyclic but have

Q: 4. Let H be a subgroup of a group G. Show that exactly one left coset of H is a subgroup.

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Q: Suppose that G is a finite simple group and contains subgroups Hand K such that |G:H| and |G:K| are…

A: Consider the finite simple group G that has subgroup H and K. |G: H| and |G: K| are relatively…

Q: * Let H be a proper subgroup of the group (Z/n)*. Prove that there are infinitely many prime numbers…

A: Given H is subgroup of Z/n* Assume there exist q1, q2 with 0≤q1,q2≤1 and q1≠q2 such that q1+Z=q2+Z…

Q: Suppose H, and H2 subgroups of the group G. Prove hat H1 N Hzis a sub-group of G. are

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

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Q: 28) Let N be a finite subgroup of a group G and assume N = for some subset S of G. Prove that an…

A: Given that N is a finite subgroup of a group G and suppose N=S for some subset S of G To prove that…

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: Suppose that G is a group and |G| = pnm, where p is prime and p > 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: If G is a cyclic group, prove for subgroup N that G is a cyclic N

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Q: Let a be an element of a group G such that Ord(a) = 32. If H is a normal subgroup of G, then Ord(aH)…

A: Result: Let G be a group and H be a normal subgroup of G. Let 'a' be an element of G such that order…

Q: e subgroups

A: Introduction: A nonempty subset H of a group G is a subgroup of G if and only if H is a group under…

Q: 2) Let (G, *) be a group and H, K be subgroups in G. Prove that subset H * K is a subgroup if and…

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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)…

A:

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…