Label each of the following statements as either true or false. If every nontrivial subgroup of a group G is a p-group, then G must be a p-group.

Q: Show that if H and K are subgroups of an abelian group G, then {hk|h € H and k e K} is a subgroup of…

A: A set G is called a group if it satisfies four properties  Closure property:  ab∈G  where a,b ∈G…

Q: If g and h are elements from a group, prove that ΦgΦh = Φgh

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Q: Label each of the following statements as either true or false   The order of an element of a finite…

A: Theorem :                           In a finite group, the order of a group element…

Q: Prove that the group G = [a, b

A: Given, the group G=a, b with the defining set of relations…

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: For any subset H of a group G, if ab^-1 is in H for all a,b element of H then H is a subgroup of G.…

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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: that a and xax^-1 have same order for all x belongs G.

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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: 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: 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: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group {e, a, b, a…

A: Given :              A group of order 4  Corollary :                    The order of an element of a…

Q: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if and only if a = a-1.

A: Let G be a group with respect to * . Let e be the identity element,and a is non identity element.…

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: 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: Label each of the following statements as either true or false. .Let H be any subgroup of a group G.…

A: Consider the given statement "Let H be any subgroup of a group G. Then H is a left coset of H in G".…

Q: In group theory (abstract algebra), is there a special name given either to the group, or the…

A: Yes, there is a special name given either to the group, or the elements themselves, if x2=e for all…

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: Prove that the trivial representation of a nite group G is faithful if and only if G = {1G}

A: To Prove: The trivial representation of the nite group G is faithful if and only if G =1G

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

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

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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: Let G be a group, and let xeG. How are o(x) and o(x) related? Prove your assertion

A: According to the given conditions:

Q: Label each of the following statements as either true or false.   The identity element in a group G…

A: In a group, the identity element is its own inverse.

Q: Let H be a subgroup of a group G and a, bEG. Then bE aH if and only if * O None of these O ab e H O…

A: here option (c) is true.

Q: Label eachof the following statement/s as either true or false.   The symmetric group Sn on n…

A: Given: The symmetric group Sn on n elements has order n. To label: The given statement is true or…

Q: Prove that any two Sylow p-subgroups of a group G are conjugate.

A: To prove that any two sylowp-subgroup of a group G are conjugate

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: Prove that if G is a finite group, then the index of Z(G) cannot be prime.

A:

Q: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

Q: Label each of the following statements as either true or false.   Every subgroup of a p-group is a…

A: Whether the statement, "Every subgroup of a p-group is p-group" is        true…

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: Prove that if every non-identity element of a group G is of order 2,then G is abelion

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Q: Let G be a group and let A1, A2, B1, B2 be the proper nontrivial subgroups of G. Suppose that A1×A2…

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Q: Let H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.

A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…

Q: If G is a cyclic group of order n, prove that for every element a in G,an = e.

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Q: Prove that if B is a subgroup of G then the coset produced by multiplying every element of B with X…

A: Solution: Let us consider (G, .) be a group and B be a subgroup of the group G. Now for any g∈G, the…

Q: (A) Prove that, every group of prime order is cyclic.

A: Let, G be a group of prime order. That is: |G|=p where p is a prime number.

Q: Label the following statement as either true or false. Every finite group G of order n is isomorphic…

A: True Cayley's Theorem states that every group G is isomorphic to a subgroup of the group T(S) of all…

Q: If a subgroup H of a group G is cyclic, then G must be cyclic. Select one: O True O False

A: we will give the counter example in support of our answer.

Q: Let G be an infinite cyclic group. Prove that G (Z,+)

A: To show that any infinite cyclic group is isomorphic to the additive group of integers

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: . Prove that the group Zm × Zn is cyclic and isomorphic to Zmn if and only if (m, n) = 1.

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

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Q: Label each of the following statements as either true or false.   The Cayley table for a group will…

A: We know that Cayley table for a group will always be symmetric with respect to the diagonal from…

Q: Let G and H be groups, and let ø:G-> H be a group homomorphism. For xeG, prove that )x).

A: Given:G and H be the groups.

Q: Prove that every group of order 375 has a subgroup of order 15.

A: According to the given information it is required to prove that every group of order 375 has a…

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: