Let G be a nonabelian group. If H and K are cyclic subgroups of G, does it follow that H ∩ k is also a cyclic subgroup?Prove that it does,or provide a counterexam

Q: Prove the following statements or give counter examples. Every characteristic subgroup is fully…

A:

Q: Let G be an Abelian group and let H 5 {g E G | |g| divides 12}.Prove that H is a subgroup of G. Is…

A: Given: G is an abelian group and H=g∈G   g divides 12. To Prove: H is a subgroup of G. And, check…

Q: every group of order 720 has a subgroup of order 12. disprove by providing counterexample

A: The statement  provided is incorrect. Not every group of order 720 necessarily has a subgroup of…

Q: Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that NG(H)…

A: " Let G be a group and H ≤ G.The subgroup H is normal in its normalizer NG(H), this imply that NG(H)…

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 in F7 the cyclic subgroup generated by x is a normalsubgroup.

A: Given: F7We need to prove that the cyclic subgroup generated by x in F7 is a normal subgroup.

Q: Prove that if G is a group with the property that the square of everyelement is the identity, then G…

A:

Q: Let G be a cyclic group and H be a subgroup of G. Prove that H is a normal subgroup in G and G/H is…

A: Given:- H be a normal subgroup of a group  G⇒gHg-1=H∀g∈G Also given GH  is a cyclic and we need to…

Q: Prove that Z ⨁ Z is not cyclic. Does your proof work for Z ⨁ Gwhere G is any group with more than…

A: To Do: Z⊕Z is not cyclic. Does the proof work for Z⊕G where G is any group with more than one…

Q: The following statements are made about (right) cosets of a subgroup H of a group G. Say whether…

A:

Q: Let H and K be subgroups of a group G. Show that their intersection H ∩ K is also a subgroup of G.…

A:

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: Let G be a group and let H and K be subgroups of G. Prove that the intersection of H and K, denoted…

A:

Q: Let G be a group. Prove the following. (a) For any i ∈ G, i and i's inverse have the same order. (b)…

A:

Q: . Let Sn be the symmetric group on n elements, let An Sn be the alternating subgroup, consisting of…

A:

Q: Assume that G is an abelian group. Let H C G be a subgroup. Prove that the factor group G/H is…

A:

Q: Let G be a group and suppose that G only has finitely many distinct subgroups. Show that |G| < ∞.

A: If possible  let G is of infinite  order .In particular let  G = {a1,a2,...,an,...} .Let us consider…

Q: Let H be a subgroup of a group G with the property that for all a andb in G, aHbH = abH. Prove that…

A:

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…

A:

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

A:

Q: {hk | h ∈ H, k ∈ K}}

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

Q: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A

A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.

Q: Let G = {a +b 2 ∈ ℝ │ a, b ∈ ℚ }. Prove that the nonzero elements of G form a group under…

A: Note : In the question, it has to be a + b2 ∈ ℝ instead of a + b 2 ∈ ℝ. So, we are solving the…

Q: Let G be an Abelian group and H 5 {x ∊ G | |x| is 1 or even}. Givean example to show that H need not…

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: Suppose that G is a finite group and let H and K be subgroups of G. Prove that |HK| = |H||K|/|HN K|.

A:

Q: Let p and q be distinct prime numbers and set n = pq. Find the number of generators of the cyclic…

A: