Let U10 be the group of units of Z10. Determine if it 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.
Determine whether the multiplicative group Z∗24 is cyclic or not. Show work to justify your answer.
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)
do problems 2.25, 2.28 and
2.29
Problem 3.25. Provide an example of a group G and subgroups H and K such that HUK is not a subgroup of G Theorem 3.26. If G is a group such that H,K S G, then (HUK) S G. Moreover, (HUK) S G is the smallest subgroup containing both H and K Theorems 3.24 and 3.26 justify the use of the word "lattice" in "subgroup lattice". In general, a lattice is a partially ordered...
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?
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...
Q3 Consider the group (Z3 x Z3, +), where again Z3 x Z3 = {(a,b) a, b e Z3}, and we define addition by (a, b) + (c,d) = (a + cb+d). So, for instance, (1, 2) + (0,2)=(1,1). (a) List all of the subgroups of (Z3 X Z3, +). You should explain why your list is complete (i.e., why there are no subgroups other than the ones you have written). (b) Which of these subgroups are cyclic? You do...
(Abstract Algebra) Please answer a-d clearly. Show your work and
explain your answer.
(a) Let G be a group of order 4 with identity e. Show that G is either cyclic or a2-e for all (b) Does the result of part (a) generalize to groups of order p2 for any positive integer p? In other words, is it the case that if G is a group of order p2 with identity e, then is either cyclic or a- e for...
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, +).
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]