The University of Sydney MATH2008 Introduction to Modern Algebra
2 The University of Sydney (iv) The elements that generate G are a, a3, a5 and a7 (see Part (iii)). MATH2008 Introduction to Modern Algebra (v) Let W = {a, a3, a5, a7}. Then W a = { wa | w ∈ W } = {a2, a4, a6, e}, (http://www.maths.usyd.edu.au/u/UG/IM/MATH2008/) and W a2 = {a3, a5, a7, a} = W . Thus W = W e = W a2 = W a4 = W a6, and W 6= W a = W a3 = W a5 = W a7. There are exactly two distinct Semester 2, 2003 Lecturer: R. Howlett right translates of W , and they have no elements in common. The set W a is a subgroup—it is the cyclic subgroup generated by a2—and W and W a are the cosets of this subgroup. Tutorial 7 (vi) The translates of {a, a5} are itself, {a2, a6}, {a3, a7} and {a4, e}. They are all disjoint from one another, and they are the costs of the subgroup 1. Let G be the cyclic group generated by an element a of order 8. {e, a4}. The translates of {a, a3, a5} are itself, {a2, a4, a6}, {a3, a5, a7}, 4 6 5 7 6 2 7 3 2 4 (i) Write down the distinct elements of G. What is the order of G? {a , a , e}, {a , a , a}, {a , e, a }, {a , a, a } and {e, a , a }. They are not the cosets of a subgroup. It is possible to find two of these translates (ii) Determine the order of each element of G. which have nonempty intersection; indeed, each element of G lies in (iii) Check that, in this group, any two elements that have the same order three distinct translates.
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