What is the index of the cyclic subgroup generated by (8,24) in the group Z12 x Z30?
What is the index of the cyclic subgroup generated by (8,24) in the group Z12 x...
Exercise 4. Consider the permutation group S7. a. Show that the subgroup generated by the element (1,2,3,4,5,6) is a cyclic group of order 6. b. Show that the subgroup generated by the element (1,3, 4, 5, 6, 7) is a cyclic group of order 6. c. Show that the subgroup generated by the element (1,2,3) is a cyclic group of order 3. d. Show that the subgroup generated by the element (6, 7) is a cyclic group of order 2....
8. (20 points) Let G Zs x Zg and let H be the cyclic subgroup generated by (3, 3). (a) Find the order of H (b) Find the orders of g = (1,1) + H, h = (1,0) + H and k = (0,1) + H in G/H (c) Classify the factor group G/H according to the fundamental theorem of finitely generated abelian groups. 8. (20 points) Let G Zs x Zg and let H be the cyclic subgroup generated...
(a) Let be a cyclic group of order . Prove that for every divisor of there is a subgroup of having order . (b) Characterize all factor groups of
1-5 theorem, state it. Define all terms, e.g., a cyclic group is generated by a single use a element. T encourage you to work together. If you find any errors, correct them and work the problem 1. Let G be the group of nonzero complex numbers under multiplication and let H-(x e G 1. (Recall that la + bil-b.) Give a geometric description of the cosets of H. Suppose K is a proper subgroup of H is a proper subgroup...
In Exercises 22 through 24, find all subgroups of the given group, and draw the subgroup diagram for the subgroups. 22. Z12 24. Z8 23. Z36
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)
Answer Question 5 . Name: 1. Prove that if N is a subgroup of index 2 in a group G, then N is normal in G 2. Let N < SI consists of all those permutations ơ such that o(4)-4. Is N nonnal in sa? 3. Let G be a finite group and H a subgroup of G of order . If H is the only subgroup of G of order n, then is normal in G 4. Let G...
Let G be a group of order 35. Show that every non-trivial subgroup of G is cyclic.
(6 points) Let G be a group of order 35. Show that every non-trivial subgroup of G is cyclic.
1. Give an example of a group, G, and a proper subgroup, H, where H has finite index in G and H has infinite order 2. Give an example of a group, G, and a proper subgroup, H, where H has infinite index in G and H has finite order. (Hint: you won't be able to find this with the groups that we work a lot with. Try looking in SO2(R)) 1. Give an example of a group, G, and...