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

Q: Let G be a group and let H and K be subgroups of G so that H is not contained in K and K is not…

A: Given that G be a group and H and K are two subgroup s.t H is not contained in K and  K is not…

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 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 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: 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: 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: Assume that (G, ) is a group and that (H, ) and (K, ) are subgroups of (G,*). Prove that (HnK,*) is…

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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: Prove that the intersection of two subgroups of a group G is a subgroup of G.

A: We will prove the statement.

Q: 32. If H and K are subgroups of G, show that Hn K is a subgroup of G. (Can you see that the same…

A: To show:

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 H be a subgroup of a group G and a, be G. Then be aH if and only if None of these O ab e H O…

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

Q: If a is an element of order 8 of a group G, and

A: Let G be a group. Let a is an element of order 8 of group G. That is, a8=e          where e is an…

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: Suppose H and K be subgroups of a finite group G with |G : H| = m and |G : K| = n. Prove that…

A: We use here, Tower law of subgroup which states that Let (G,∘) be a group.  Let H be a subgroup of G…

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

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: The center of G is a commutative subgroup of G.

A: To prove that the center of G is a commutative subgroup of G.

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: Prove that if G is a finite cyclic group, Hand K are subgroups of G, and H =f. K, then IHI =f. IKI

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Q: If G is a finite group, H ≤ G, the order of H divides the order of G: | H | / | G | Prove

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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…

A:

Q: prove That :- let H and K be subgroups of agroupG of the m is ormal a HK is subgroup of G →1f one

A: Subgroup of a group G

Q: Let H  and K  be subgroups of a group G with operation * .  Prove that HK  .is closed  under the…

A: Given information: H  and K  be subgroups of a group G with operation * To prove that HK is a closed…

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: If N is a subgroup of an Abelian group, prove that is Abelian. N |

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Q: If H and K are subgroups of a group G, prove that ANB is a subgroup of G.

A: GIVEN if H and K are the subgroup of a G, prove that  A∩B  is a subgroup of G

Q: Let G be a group, H4G, and K < G. Prove that HK is a subgroup of G. Bonus: If in addition K 4G,…

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

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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: 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: If G is a group and g E G, the centralizer of g E G, is the set CG(g) := {a E G : ag =ga} that is,…

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Q: Although (H,*) and (K,*) are subgroup of a group (G,*) then (H * K, ) may field to be subgroup of…

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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: 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: Prove that a group that has more than one subgroup of order 5 musthave order at least 25.

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Q: Prove or disprove this statement. If G is a group in which every proper subgroup is cyclic, then G…

A: using the concept of cyclic group and subgroup...

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: A subset of H of a group G is a subgroup of G if the operation on G makes H into a group. Prove that…

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Q: If H and K are subgroups of a group G then H n K is a subgroup of G.

A: Note:  according to our guidelines we can answer first question and rest can be reposted.   Lemma:…

Q: Prove that the intersection of two subgroups of G is itself a subgroup of G.

A: Suppose that, H1 and H2 be the sub groups of G. Also, let H=H1∩H2 . Consider one step group test:…

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: Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G is cyclic.

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Q: Although (H,*) and (K,*) are subgroup of a group (G,) then (H * K,*) may field to be subgroup of (G,…

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Q: (a) If G is abelian and A and B are subgroups of G, prove that AB is a subgroup of G. (b) Give an…

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Q: 5. If H. aEA are a family of subgroups of the group G, show that is a subgroup of G.

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Q: Prove that ifH and K are subgroups of a group G with operation *, Question 8. then HNK is a subgroup…

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Q: i have included a picture of the question i need help understanding.thank you in advance. please…

A: Let H and K are two subgroups of the group G.To show