Let G be a group that has proper subgroups of order 6, 8 and 12. What...
19. Let G be a group with no nontrivial proper subgroups. (a) Show that G must be cyclic. (b) What can you say about the order of G?
Question 4 a) If a group G has order 323, what are the possible orders of its proper b) Let G<a > be a cyclic group of order 10. Is the map f: G-> G define«d subgroups: by Зі i. f(a) - a and f(a')-a agroup isomorphism? ii(a) and fa a a group isomorphism? 12, 5, 5
Let G be a cyclic group of order 30. Find all the subgroups of G. Write the lattice of subgroups. Justif your answer and cite the theorems that allow you to determine such lattice.
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...
(8) (5 Points Total) Let (G, ) be a group and H and K are subgroups of G, such that H¢K and K¢ H. Is HuK is a subgroup of G? Prove or disprove. (Note: it is not enough to say Yes or No) Answer:
(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.
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...
Let G be a finite group, and let H be a M be a subgroup of G such that H C M C G. What are the possible orders for M? Why? Let G possible orders of subgroups of S5 which contain D5? subgroup of G. Finally, let S5, and let H = D5. What are the _ Let G be a finite group, and let H be a M be a subgroup of G such that H C M...
Please answer all the four subquestions. Thank you! 2. In this problem, we will prove the following result: fG is a group of order 35, then G is isomorphic to Z3 We will proceed by contrd cuon, so throughout the ollowing questions assume hat s grou o or ㎢ 3 hat s not cyc ić. M os hese uuestions can bc le nuc endent 1. Show that every element of G except the identity has order 5 or 7. Let...
19. Let H and K be subgroups of a group G, where H = 9, 1K1 =12 and where the index [G:HNK] =IGI. Find (HNKI.