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

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: 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 G be a group and a E G be a certain fixed element of G. The centralizer of a in G is C(a) = {g €…

A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…

Q: 3. Let G be a group, H a subgroup of G, and g and element of G. (a) Prove that {ghg-1 | h e H} is…

A: Let G be a group, H is a subgroup of G, and g be an element of G

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: For any group G, Z(G) ≤ [G, G]. Prove or give a counter example

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

A: Given that H and K are the finite subgroups of a group G and also an element a such that a∈G Here,…

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

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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: Let G be a group, let H be a proper subgroup of G, and let a E G. Prove that the left coset aH is a…

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Q: Show that if G is a finite group of even order, then there is an a EG such that a is not the…

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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: 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 finite group, p prime, and Ha Sylow p-group. Prove that H is normal in G if and only if H…

A: Given, G is a finite group and H is a sylow p-group o(G)=n, where n is a finite number. p/o(G) from…

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

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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: Let G be a group. The centre of G is defined as the set Z(G) = {g ∈ G : gx = xg for all x ∈ G}.…

A: Let G be a group. The center ZG of group G is defined as follows. ZG = g∈G : gx = xg , ∀x∈G We need…

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: Let G be a group and a e G. Show that o(a) = o(a-). order n, then ba also has order n.

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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 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: Given the groups R∗ and Z, let G = R∗ ×Z. Define a binary operation ◦ on G by (a, m) ◦ (b, n) = (ab,…

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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: {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 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: 2. Let G be a group. Prove or disprove that Z= {x E G: xg= gx for all g€ G} Isa Subgroup of G.

A: To show Z is a subgroup of G, we need to show that (a) Z is non empty (b) For every a , b∈Z , we…

Q: For any group G, GIZ(G) is isomorphic to Inn(G)

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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 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: 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: Let G be a group with identity element e, and let H and K be subgroups of G. Assume that (i) H and K…

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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…

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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 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: 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: Suppose that G is a group and |G| = pnm, where p is prime and p > m. Prove that a Sylow…

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Q: Let be a group and Ha normal subgroup of G. Show that if y.VEG such that xyEH then yx EH

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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: Let G be a group and let a E Ga G with a = 8. the order of a² is not equal to the order of the…

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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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