19. Let G be a group with no nontrivial proper subgroups. (a) Show that G must...
Let G be a group that has proper subgroups of order 6, 8 and 12. What is the least possible order of G.
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...
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 show step by step solution with a clear explanation!
2. Let G be a group of order 21. Use Lagrange's Theorem or its consequences discussed in class to solve the following problems: (a) List all the possible orders of subgroups of G. (Don't forget the trivial subgroups.) (b) Show that every proper subgroup of G is cyclic. (c) List all the possible orders of elements of G? (Don't forget the identity element.) (d) Assume that G is abelian, so...
14. Suppose that G is a group in which {1} and G are the only subgroups. Show that G is finite and, in fact, is cyclic of order 1 or a prime.
4. (a) (3 points) List all the subgroups of the symmetric group S3. (b) (4 points) List all the normal subgroups of Sz. (c) (3 points) Show that the quotient of S3 by any nontrivial normal subgroup is a cyclic group.
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.
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
Abstract Algebra 1 a) Prove that if G is a cyclic group of prime order than G has exactly two subgroups. What are they? 1 b) Let G be a group and H a subgroup of G. Let x ∈ G. Proof that if for a, b ∈ H and ax = b then x ∈ H. (If you use any group axioms, show them)