If a subgroup H of a group G is cyclic, then G must be cyclic. Select one: O True O False
Q: Show that if H and K are subgroups of G then so is H ∩ K.
A: Given that H and K are subgroup of group G. We have to show that H∩K is a subgroup of group G.…
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…
A:
Q: ) If G is a group and X is a G-set, then the subset (g in G: for all x
A: To explain the given statement is true or false as,
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: Suppose that G is a cyclic group such that Ord(G) = 48. The number of subgroups that G has is * O 8…
A: Q1. Third option is correct. Q2. Second option is correct.
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: 5. Let p and q be two prime numbers, and let G be a group of order pq. Show that every proper…
A: We have to prove that: Every proper subgroup of G is cyclic. Where order of G is pq and p , q are…
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: Theorem: Let (K,+)is a subgroup of a group (H, ) and (H,) is a subgroup of a group (G,) then (K, )is…
A:
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: A nonempty subset of a group, that is closed under the operation of the group, is a subgroup. Birini…
A: A nonempty subset of a group is a subgroup only if it is a group under the same binary operation.
Q: If H is a normal subgroup of a finite group G and |H| = pk for someprime p, show that H is contained…
A: H is a normal subgroup of a finite group G and |H| = pk for some prime p.
Q: Suppose that K is a proper subgroup of H and H is proper subgroup of G. If |K| = 42 and |G| = 420,…
A:
Q: It is not possible that, for a group G and H and K are nomal subgroups of G, H is isomorphic to K…
A: Let G be a group and H and K are normal subgroups of G
Q: In Z, find all generators of the subgroup <3>. If a has infinite order,find all generators of…
A: Since a has infinite order, the same holds for a 3 since if it would have order n < ∞, then 1 =…
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: If N is a normal subgroup of a group G, and if every member of N and G/N have a finite order, prove…
A: Given: If N is a normal subgroup of a group G, and if every member of N and GN have a finite order…
Q: f H and K are two subgroups of a group G, then show that for any a, b ∈ G, either Ha ∩ Kb = ∅ or Ha…
A: If H and K are two subgroups of a group G, then show that for any a, b ∈ G,either Ha ∩ Kb = ∅ or Ha…
Q: Suppose that in the definition of a group G, the condition that there exists an element e with the…
A: We need to prove that;
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: 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: Although (H,*) and (K,*) are subgroup of a group (G,*) then (H * K, ) may field to be subgroup of…
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Q: Use the definition of a normal subroup to prove Proposition 2.3.7: IfGis an Abelian group, then…
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Q: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10…
A: If G is cyclic group and order of G is 'n'. Then number of subgroups of G is equal to number of…
Q: Show that if p and q are distinct primes, then the group ℤp × ℤq is isomorphic to the cyclic group…
A: We have to show that if p and q are distinct primes, then the group Zp×Zq is isomorphic to the…
Q: Every finite group is cyclic
A:
Q: The identity element in a subgroup H of a group G must be the same as the identity element in G…
A: The identity element in a subgroup H of a group G must be the same as the identity element in G.
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: 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: 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…
A:
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: If G is a cyclic group, prove for subgroup N that G is a cyclic N
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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: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10 O…
A:
Q: Although (H,*) and (K,*) are subgroup of a group (G,) then (H * K,*) may field to be subgroup of (G,…
A:
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: 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: : Show that in a group G, if a? = e,Vx E G, then G is a commutative. %3D
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Q: Let H be a subgroup of a group G and a, bEG. ThenbE aH if and only if * O None of these O ab EH O…
A:
Q: Suppose that G is a group such that Ord(G) = 36. The number of subgroups %3D that G has is 4 О 12 О…
A: Order of a group: Let G be a group and n be the number of elements in the group. Then, order of…
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- Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?43. Suppose that is a nonempty subset of a group . Prove that is a subgroup of if and only if for all and .34. Suppose that and are subgroups of the group . Prove that is a subgroup of .