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

Q: Let G be a finite group of order n. Let g be an element of G. Prove that gn=e.

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Q: Let (G,*) be an a belian group, if (H,) and (K,*) are subgroup of (G,*) then (H * K,*) is a subgroup…

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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 G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G. Prove that if…

A: Given that, Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G.

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: Let H and K be subgroups of a finite group G with H C KC G. Prove that |G:HI |G:K| |K:H].

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Q: Let G be a group and g E G. Show that Ø:Z→G by Ø(n) = g" is a homomorphism and isomorphism.

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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: 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 ø be a homomorphism from a group G to a group H. Let K be a subgroup of H. Prove that ø (K) = {…

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Q: Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

A: Given that Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

Q: Let G be a group and a e G. Prove that C(a) is a subgroup of G. Furthermore, prove that Z(G) = NaeG…

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

A: Given: G is a group with order 187. Theorems used: Any group of prime order is cyclic Lagrange…

Q: let G be a group of order p^2 where p is prime. Show that every subgroup of G is either cyclic or…

A: Given that G is a group of order p2, where p is prime.To prove that every subgroup of G is either…

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

A: Given G is finite group and we have to prove G is a p-Group of and only if |G| is a power of p.

Q: H be a subgroup of G.

A: We have to find out the truth value of the given statements. It is given that H is a subgroup of G.…

Q: Let (G,*) be an a belian group, if (H,*) and (K,*) are subgroup of (G,*) then (H * K,*) is a…

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Q: 5. Suppose G is a group of order 8. Prove that G must have a subgroup of order 2.

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

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Q: a. Prove or Disprove. If H and K be normal subgroups of a group G and H is isomorphic to K, then G/H…

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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 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: Show that if aH=H then a belongs to H. H is a subgroup of a group G and a is an element of G

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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: Give an example of a finite group G with two normal subgroups H and K such that G/H = G/K but H 7 K.

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Q: Give an example of subgroups H and K of a group G such that HKis not a subgroup of G.

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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: 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: 2. Let G be a group and let H be a subgroup of G. Define N(H) = { x = G | xHx™¹ = H}. Prove that…

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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 with |G|=221 then every proper subgroup of G is: * О Сyclic Non abelian O Non…

A: We are given that G is a group with G=221 then we have to tell about the every proper subgroup of G.

Q: Let let G N Subgroup be be of G a a group and normal of finite

A: To prove that H is contained in N, we first prove this: Lemma: Let G be a group.H⊂G. Suppose, x be…

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. Prove that Z(G) < G.

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Q: Let H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.

A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…

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: 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 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: 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 G be a group with G|=221 then every proper subgroup of G is: " O Non cyclic O None of these O…

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Q: Let G be a group with order n, with n> 2. Prove that G has an element of prime order.

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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: Let G and H be groups, and let ø:G-> H be a group homomorphism. For xeG, prove that )x).

A: Given:G and H be the groups.

Q: 1. Let G be a group and let H, H, .. H, be the subgroups of G. The ...

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