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

Q: Prove or give counterexample. Let G be a group and H ≤ G. The subgroup H is normal in its…

A:

Q: Prove that a group of order 105 contains a subgroup of order 35.

A: Let G be a group of order 105. We need to show that G contains a subgroup of order 35. We know the…

Q: Prove the next statement: If H is a subgroup of a finite abelian group G, then G has a group…

A: Introduction: In mathematics, an isomorphism is a type of mapping or function between two…

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: Let H be a proper subgroup of a finite group G. Show that G is notthe union of all conjugates of H.

A: Let H be a proper subgroup of a finite group G. We how prove that G is not the union of all…

Q: let x and g be elements of a group G. show that x and gxg-1 have the same order. now prove that xy…

A: Hence, proved.Explanation:Step 1:To prove that x and gxg-1  have the same order for any elements x…

Q: Let G = V×Z3 and let H be the subgroup (a)×(2) of G. Calculate “. (The quotient group itself, not…

A:

Q: Let G and H be groups. Let p : G → H be a homomorphism and let E < H be a subgroup. Prove that p(E)…

A: Given: φ:G→H is a group homomorphism and E≤H. To prove: a) φ-1(E)≤G b) If E ⊲ H then φ-1E ⊲ G

Q: to a subgroup of Sn for n ≥ 3.

A:

Q: Let a and b be elements of a group.  If |a| and |b| are relatively prime, show that intersects =…

A: Let m and n be the order of the elements a and b of a group G. Given that the orders of a and b are…

Q: 2. Let G and H be groups, and let a: G H be a homomorphism. Prove the following statements. (a) a(G)…

A: Let us prove above results

Q: Let G be a group and let x ∈ G. Prove that if a has order of 12, then x3 has order 4.

A:

Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: Prove that S, is isomorphic to a subgroup of An+2-

A: An even permutation can be obtained as the composition of an even number and only an even number of…

Q: Suppose the class equation of a group G is |G| = 1 + 4 + 5 + 5 + 5. Show that G has an element of…

A:

Q: (6) Let G be a noncommutative group. Show that G has a nontrivial subgroup.

A: We recal Cauchy's theorem and then use it

Q: In S3, find an example of an H < G, and g1, g2 ∈ G such that the two conjugate subgroups are not…

A: Let H be a subgroup of G Then a H1={ghg | g,h belongs to G,H respectively}. is a conjugate subgroup

Q: If G and H are groups with identities 16 and ¹H, respectively, and f:G ---> H is a homomorphism,…

A:

Q: Let G be a finite group. Let xeG, and let i>0. Then prove that o(x) gcd(i,0(x))

A: To prove the required identity on the order of the group element

Q: Prove that If |G|=np with p>n and p prime, and H is a subgroup of G with order p, then: H is normal…

A: We proved by contradiction that H is normal in G  The procedure is given below