Question 4 a) If a group G has order 323, what are the possible orders of...
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...
Let G be a group that has proper subgroups of order 6, 8 and 12. What is the least possible order of 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...
3 Let p and q be prime numbers and let G be a non-cyclic group of order pq. Let H be a subgroup of G.Show that either H is cyclic or H-G. 12 - Let I and J, be ideals in R. In, the homomorphismJ f: (!+J a → a+J use the First Isomorphism Theorem to prove that I+J
f. In Qlrj the ged of ()-4 and ()1 is g. The polynomial h(x) with real coefficients has the complex root 4 -5i. A second root of h(x) is h. Consider the tower of fields: K> E> F with [K : F 54 and [K:E] =6. Then [E:f]= i. If f(x) r6245r62330650 is divided by +5, the remainder is j. The number of subgroups of a group G having order 73 is
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...
Let G = {1, 3, 5, 9, 11, 13} and let represent the binary operation of multiplication modulo 14. (a) Prove that (G, ) is a group. (You may assume that multiplication is associative.) (b) List the cyclic subgroups of (G, ). (c) Explain why (G, ) is not isomorphic to the symmetric group S3. (d) State an isomorphism between (G, ) and (Z6, +).
4. Suppose G is a group of order n < 0. Show that if G contains a group element of order n, then G is cyclic.
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...
2. Let G {g, g. . . , gn-le} be a cyclic group of order n, H a group, and h є H. Define a function φ : G → H by φ(gi-hi for all 0 < i n-1. Show that φ is a group homomor- phism if and only if o(h) divides o(g). Warning: mind your modular arithmetic! [10]