U(13) has six cyclic subgroups. List them.
Order and Cyclic Subgroups: Problem 5 Previous Problem Problem List Next Problem (1 point) Let x be an element of order 91 in a group G (not necessarily cyclic, finite, or Abelian). How many distinct subgroups of G are contained in (x)?
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.
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.
Consider the additive group ℤ(20). (a) How many subgroups does ℤ(20) have? List all the subgroups. For each of them, give at least one generator. (b) Describe the subgroup < 2 > ∩ < 5 > (give all the elements, order of the group, and a generator). (c) Describe the subgroup <2, 5> (give all the elements, order of the group, and a generator).
In a pool 210 different subgroups of six that could be produced from a pool of ten applicants. Suppose the applicant pool is made up of three women and seven men. How many of the 210 subgroups of six would contain exactly two women and four men?
4) (5 marks) List all of the subgroups of Z24. List all of the generators of Z24- 1281 E91 par jul/110101 G
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)
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, +).
21. List as many subgroups as you can of the group of symmetries of a circle.
List at least six protocols used on the Internet and briefly describe what each of them are used for.