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

Q: 1. Let a and b be elements of a group G. Prove that if a E, then C. 2. Let a and b be elements of a…

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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 G1 and G2 be groups, and let G = G1 × G2. Write e and e2 for the identities of G1 and G2…

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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 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: 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: 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: 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 H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

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, 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 G is a finite group with |G|<160 and G has subgroups of orders 1O, 16 and 20 then the order of G…

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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: 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 H be a subgroup of a group G. Prove that the set HZG) = {hz | h E H, z E Z(G)} is a subgroup…

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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: 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 with |G|<120 and G has subgroups of orders 1O, 15 and 20 then the order of G…

A: Using Lagrange's theorem it can be written that if G is a finite group and B is a subgroup of G,…

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: 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 finite group. Let E G and let xG be the conjugacy class of x. Prove that x| < |[G, G]],…

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Q: An element a of a group G has order n E z+ if and only if a" = e.

A: The given statement is False.

Q: {hk | h ∈ H, k ∈ K}}

A: We have to prove that  {hk|h∈H, k∈K} is a subgroup of G.

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: Suppose that a is an element of a group G. Prove that if there is some integer n, n notequalto 0,…

A: Suppose that a is an element of a group G. Prove that if there is some integer n, such that n≠0 and…

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

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Q: Let G be a finite group and H1, H2, ., Hk be subgroups of G. (a) Show that k H1n H2n. n Hk < G. Hi…

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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: Question 2. Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G…

A: Given: G is a group that has exactly one non-trivial proper subgroup. To prove: G is cyclic and…

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…

A: F hv

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

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Q: Let G be a group. Prove that Z(G) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called the…

Q: Let H and K be subgroups of the group G, and let a, b E G. Show that either aH n bK = Ø or else aH N…

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Q: Show that if G is a finite group with identity e and with an even number of elements, then there is…

A: Given,If G is a finite group with identity e and with an even number of elements,then there is a ≠e…

Q: dicfin et Prove that a group G has exactly 3 6. - subgroups iff G is a ylic grop ef ender på pis…

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Q: 2. Let o : G → G' be a homomorphism of groups, and let H be a subgroup of G. (a) Let a E G. Prove…

A: Given   ϕ:G -&gt; G' is a homomorphism. H is a subgroup of G. The solution to the subparts is given…

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: 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 H and K be subgroups of a group G and assume |G : H| = +0. Show that |K Kn H |G HI if and only…

A: Given:

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 G is a finite group with |G|<180 andG has subgroups of orders 10, 18 and 30 then the order of G…

A: Use the fact that order of subgroup divides order of group

Q: Let K and H be subgroups of a finite group G with KCHCG.lf [G:K] = 12 and [H:K] = 3. Then, [G:H] =…

A: Let , K  and  H  be subgroups of finite group G. Also . K ⊆ H ⊆ G Here , G : K = 12  , H : K = 3 We…

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: 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 G, H, and K be finitely generated abelian groups. Prove or disprove: If G × H ∼= K × G, then H…

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