Consider the group G = ℤ9 ⨉ ℤ27 where the operation is done componentwise, that is (a, b) * (c, d) = (a +9 c, b +27 d), for all (a, b), (c, d) in G. (a) Find the elements of the subgroup <(1, 3)>. (b) Compute [G : <(1, 3)>]. (c) Find the elements of the left coset (1, 6) + <(1, 3)>.

Consider the group G = ℤ9 ⨉ ℤ27 where the operation is done componentwise, that is (a, b) * (c, d) = (a +9 c, b +27 d), for all (a, b), (c, d) in G. (a) Find the elements of the subgroup <(1, 3)>. (b) Compute [G : <(1, 3)>]. (c) Find the elements of the left coset (1, 6) + <(1, 3)>.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 3E: 3. Consider the group under addition. List all the elements of the subgroup, and state its order.

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Question

Consider the group G = ℤ₉ ⨉ ℤ₂₇ where the operation is done componentwise, that is
(a, b) * (c, d) = (a +₉ c, b +₂₇ d), for all (a, b), (c, d) in G.

(a) Find the elements of the subgroup <(1, 3)>.
(b) Compute [G : <(1, 3)>].
(c) Find the elements of the left coset (1, 6) + <(1, 3)>.

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