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

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 H be a subgroup of a group G and a, bEG. Then be aH if and only if * a-1b e H O None of these…

A: Given H is a subgroup of G. We need to find a necessary and sufficient condition for a belongs to…

Q: Let G be a group and H, K are subgroups of G with HK=KH. Prove that HK is a subgroup of G.

A: Given that, G be a group and H, K are sub groups of G with HK=KH. Let x∈HK. Then x=hk for some…

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: Let H be a subgroup of a group G and a, be G. Then bE aH if and only if * O a-1b eH O ab-1 eH O None…

A: We know that  b∈bH                    (1) We know that  aH = bH if and only if a-1b ∈H…

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: Suppose that G is a group of order 168. If G has more than oneSylow 7-subgroup, exactly how many…

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Q: Let A be a subset of the group G. Prove that the normalizer of A, NG(A) = {g e G: gAg=A }, is a…

A: Consider the provided question, According to you we have to solve only question no. 2. (2)

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: Let H be a subgroup of a group G and a, b E G. Then be aH if and only if *

A: So,  a, b  belongs to H, and we have  b∈aH  Hence,  b = ah     -- for some element of H  Hence,  a-1…

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: Suppose H is a distant and normal subgroup of a group G. Prove that each subgroup of H is a normal…

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Q: Let G be an abelian group and suppose that H and K are subgroups of G. Show that the following set…

A: According to the given information, Let G be an abelian group and suppose that H and K are subgroups…

Q: Let G be a finite group and H a subgroup of G of order n. If H is the only subgroup of G of order n,…

A: Given, G is a finite group and H is a subgroup of G of order n.

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 and H, KG normal subgroups of G. Prove HnK≤ G.

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Q: Let G be a group, prove that the center Z(G) of a group G is a normal subgroup of G.

A: Let G be a group. Consider the subgroup ZG=x∈G | ax=xa.

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: If G is a finite group with |Gl<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A: Given: The group G is a finite group with | G | &lt; 120 and G has subgroups of orders 10, 15, and…

Q: Let G be a group and let H and K subgroup of G. Prove that the intersection H and K is a subgroup of…

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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 G be a group and H a subgroup of G. If [G: H] = 2 then H ⊲ G, where [G: H] represents the index…

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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: If a simple group G has a subgroup K that is a normal subgroup oftwo distinct maximal subgroups,…

A: Here given G is simple group and K is a normal subgroup of G. Then use the definition of simple…

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…

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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 two subgroups of a group G. Let HK={ab|a∈H,b∈K}. Then HK is a subgroup of G. true or…

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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 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: If G is a finite group with IG|<160 and G has subgroups of orders 10, 16 and 20 then the order of G…

A: Given :- order of a group G... |G| &lt; 160 Also order of subgroups of group G are 10, 16 , 20. We…

Q: Let A be a subset of the group G. Prove that the normalizer of A, NG(A) = { g e G : gAg-1 = A}, is a…

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Q: Let H be a subgroup of a group G and a, be G. Then b E aH if and only if ab-1 e H O ab e H O None of…

A: Ans is given below

Q: 7. Let G be a group, prove that the center Z(G) of a group G is a normal subgroup of G.

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Q: Show that if G is a group of order 168 that has a normal subgroup oforder 4, then G has a normal…

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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: 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: 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 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: 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 H and K be subgroups of a finite group G. Show that |HK |HK= |HОКI where HK (hk hE H, k E K}.…

A: let D = H ∩K then D is a subgroup of k and there exist a decomposition of k into disjoint right…

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

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Q: Let H be a subgroup of G, define C(H) the centralizer of H.

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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…