Let G be a group of order pq where p and q are distinct primes andp < q. Prove that the Sylow q-subgroup is normal in G.

Q: Recall that the center of a group G is the set {x € G | xg = gx for all g e G}. Prove that he center…

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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: 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: Prove that every group of order 1225 has a normal abelian Sylow 5-subgroup.

A: Since not a particular question asked as per guidelines solution to only first question  is given…

Q: Prove that there is no simple group of order p2q, where p and q areodd primes and q &gt; p.

A: Let G be a group. |G| = p2q, where p and q are odd primes and q &gt; p. The first Sylow Theorem: A…

Q: Let G be an Abelian group and H = {x E G | |x| is odd}. Prove thatH is a subgroup of G.

A: Given: To prove H is a subgroup of G.

Q: Let N be a normal subgroup of a finite group G. Use the theorems ofthis chapter to prove that the…

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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: Find the right cosets of the subgroup H in G for H = ((1,1)) in Z2 × Z4.

A: Let, the operation is being operated with respect to dot product. Elements of ℤ2=0,1 Elements of…

Q: Let H be a subgroup of a group G and S the set of all left cosets of H in G. Show that there is a…

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

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Q: If H is a normal subgroup of G and |H| = 2, prove that H is containedin the center of G.

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Q: Let H be a subgroup of a group G, S {Hx: x e G}. %3D Then prove that there is a homomorphism ofG…

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Q: Let H be a subgroup of a group G and S the set of all left cosets of H in G. Show that there is a…

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Q: If N and M are normal subgroups of G, prove that NM is also a normalsubgroup of G.

A: Given N and M are normal N and M are normal subgroup of G. We have to prove: NM is a subgroup of G…

Q: Let H be a subgroup of a group G and S the set of all left cosets of H in G. Show that there is a…

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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: Let G = V×Z3 and let H be the subgroup (a)×(2) of G. Calculate “. (The quotient group itself, not…

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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 finite p-group of order p". Show that for all 0<kSn, there is a subgroup order p and each…

A: Given: Let G be a finite p-group of order pn. We have to prove for all 0≤k≤n there is a subgroup of…

Q: Let G be a group and H, KG normal subgroups of G. Prove HnK≤ G.

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Q: . Let H and K be normal subgroups of a group G such that HCK, show that K/H is a normal subgroup of…

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Q: Let G be a group and let H be a subgroup of G with |G : H| = 2. Prove that H a G, that is, H 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: . Let H and K be normal subgroups of a group G such nat HCK, show that K/H is a normal subgroup of…

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Q: Let M and N be normal subgroups of G. Show that MN is also a normal subgroup of G

A: It is given that M and N are normal subgroups of G. implies that,

Q: Let H and K be normal subgroups of a group G such at HCK, show that K/H is a normal subgroup of G/H.

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Q: let H be a normal subgroup of G and let a belong to G . if the element aH has order 3 in the group…

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Q: Let C be a normal subgroup of the group A and let D be a normal subgroup of the group B. Prove that…

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Q: 3) Show that the subgroup of Dg is isomorphic to V4.

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Q: Let H be a subgroup of a group G and S the set of all left cosets of H in G. Show that there is a…

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Q: Let G be an abelian group,fo f fixed positive integer n, let Gn={a£G/a=x^n for some x£G}.prove that…

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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 H be a subgroup of G of index 2. Prove that H is a normal sub-group of G.

A: the prove is given below...

Q: Let H be a subgroup of a group G and S the set of all left cosets of H in G. Show that there is a…

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Q: Let H be a subgroup of a group G and S the set of all left cosets of H in G. Show that there is a…

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Q: Let H and K be normal subgroups of a group G such that HCK, show that K/H is a normal subgroup of…

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Q: Let G be a group of order 425. Prove that if H is a normal bgroup of order 17 in G then H < Z(G).

A: Let G be a group of order 425. Prove that if H is a normal subgroup of order 17 in G. Then H≤ Z(G).

Q: Let G and H be groups. Prove that G* = {(a, e) : a E G} is a normal subgroup of G × H.

A: We atfirst show that G* is a subgroup of G×H . Then we show that G* is normal in G×H

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: Prove that if H is a normal subgroup of G of prime index p then for all K < G either (1) K < H or…

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

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Q: 6. If N< G and G/N is free, prove that there is a subgroup H such that G = HN and HoN= 1. (Use the…

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Q: a. If G is a group of order 175, show that GIH=Z, where H is a normal subgroup of G. b. Show that Z…

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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: 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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