Let G be a group of order 231 = 3 · 7 · 11. Let H,...

Question

Question

Let G be a group of order 231 = 3 · 7 · 11. Let H, K and N denote sylow 3,7 and 11-subgroups of G, respectively.

a) Prove that K, N are both proper subsets of G.

b) Prove that G = HKN.

c) Prove that N ≤ Z(G). (you may find below problem useful).

a): <|/ is a normal subgroup, i.e. K,N are normal subgroups of G

(below problem): Let G be a group, with H ≤ G and N proper subset of G.

a) Show that H acts on N by conjugation, i.e. that ∗ : H × N → N given by h ∗ g = hgh^(−1) is a group action.

b) Show that the associated permutation representation ϕ : H → $S_{N}$ restricts to a homomorphism ψ : H → Aut(N).

c) Use part b) to argue that a group of order 22 is either cyclic or isomorphic to $D_{11}$ .

Answer 1

Answer #1

Answer 2

Similar Homework Help Questions

Problem 3. Subgroups of quotient groups. Let G be a group and let H<G be a...

Problem 3. Subgroups of quotient groups. Let G be a group and let H<G be a normal subgroup. Let K be a subgroup of G that contains H. (1) Show that there is a well-defined injective homomorphism i: K/ H G /H given by i(kH) = kH. By abuse of notation, we regard K/H as being the subgroup Imi < G/H consisting of all cosets of the form KH with k EK. (2) Show that every subgroup of G/H is...
Please prove C D E F in details? 'C. Let G be a group that is...

Please prove C D E F in details? 'C. Let G be a group that is DOE smDe Follow the steps indicated below; make sure to justify all an Assuming that G is simple (hence it has no proper normal subgroups), proceed as fo of order 90, The purpose of this exercise is to show, by way of contradiction. How many Sylow 3sukgroups does G have? How many Sylow 5-subgroups does G ht lain why the intersection of any two...
4 Let G be an unknown group of order 8. By the First Sylow Theorem, G must contain a subgroup H of order 4 (a) If al...

4 Let G be an unknown group of order 8. By the First Sylow Theorem, G must contain a subgroup H of order 4 (a) If all subgroups of G of order 4 are isomorphic to V then what group must G be? Completely justify your answer. (b) Next, suppose that G has a subgroup H one of the following C Then G has a Cayley diagram like Find all possibilities for finishing the Cayley diagram. (c) Label each completed...

(10) Let G be a finite group. Prove that if H is a proper subgroup of...

(10) Let G be a finite group. Prove that if H is a proper subgroup of G, then |H| = |G|/2. (11) Let G be a group. Prove that if Hį and H2 are subgroups of G such that G= H1 U H2, then either H1 = G or H2 = G.
Let G be a group and let H,K be normal subgroups of G such that H∩K...

Let G be a group and let H,K be normal subgroups of G such that H∩K = {e} and that G = {hk|h ∈ H,k ∈ K}. (1)Prove that for every h∈H, k∈K we have kh(k^-1)(h^−1) = e in G. (2) Prove that the group G is isomorphic to H × K. Hint: For (2), consider the map φ : H ×K → G, defined as φ(h,k) = hk, whereh ∈ H,k ∈ K.
4. Let G be a group. An isomorphism : G G is called an automorphism of G. (a) Prove that the set,...

the following questions are relative,please solve them, thanks! 4. Let G be a group. An isomorphism : G G is called an automorphism of G. (a) Prove that the set, Aut(G), of all automorphisms of G forms a group under composition. (b) Let g E G. Show that the map ф9: G-+ G given by c%(z)-gZg", įs an automorphism. These are called the inner automorphisms of G (c) Show that the set of all g E G such that Og-Pe...

The Sylow theorems state the following facts about a finite group G, of order |G| =...

The Sylow theorems state the following facts about a finite group G, of order |G| = p^m (with p prime, k positive integer, and p not dividing m) a Sy1: There exist subgroups in G of size p*, called Sylow p-subgroups particular prime p, are conjugate Sy2: All Sylow p-subgroups in G, for a Sy3: The number of Sylow p-subgroups in G is congruent to 1 modulo p, and this number divides m Consider the symmetric group S9 of permutations...
quention for 8 iz) 23)1Dy ave 7. (10M) Prove that o: Z x Z Z given...

quention for 8 iz) 23)1Dy ave 7. (10M) Prove that o: Z x Z Z given by (a, b) a+b homomorphism and find its kernel. Describe the set is a 8. (10M) Prove that there is no homomorphism from Zs x Z2 onto Z4 x Z 9.(10M) Let G be a order of the element gH in G/H must divide the order of g in G. finite group and let H be a normal subgroup of G. Prove that (16M)...
Let a : G + H be a homomorphism. Which of the following statements must necessarily...

Let a : G + H be a homomorphism. Which of the following statements must necessarily be true? Check ALL answers that are necessarily true. There may be more than one correct answer. A. If kera is trivial (i.e., ker a = {eg}), then a is injective. B. If the image of a equals H, then a is injective. C. The first isomorphism theorem gives an isomorphism between the image of a and a certain quotient group. D. The first...

question for 10. (16M) Let H and K be subgroups of G. Define HK = {hk...

question for 10. (16M) Let H and K be subgroups of G. Define HK = {hk |h E H,kE K}. Suppose K is normal in G. Prove (a) HK is a subgroup of G. (b) HnK is a normal subgroup of H; K is a normal subgroup of the subgroup H K. HK K H (c) HnK (16M) Let H and K be subgroups of G. Define HK = {hk |h E H,kE K}. Suppose K is normal in G....