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MTH 461: Survey of Modern Algebra, Spring 2021 Homework 4 Homework 4 1. List the left and right cosets of the subgroups in the following list. (a) The subgroup x5y, generated by 5 pmod 20q, inside pZ5 , `q. (b) The subgroup 4Z “ t4k ∶ k P Zu inside the group pZ, `q. (c) The subgroup A3 inside the symmetric group S3 . (d) The subgroup H “ te, p12qp34q, p13qp24q, p14qp23qu in the group A4 . (e) The subgroup H “ te, p123q, p132qu in the group A4 . For which of these examples does it happen that every right coset is a left coset, and every left coset is a right coset? 2. Let G be a group and H Ă G a subgroup with index 2, i.e. rG ∶ Hs “ 2. Show that aH “ Ha for all a P G. 3. Recall that GL2 pRq is the group of real 2 ˆ 2 matrices with non-zero determinant, and SL2 pRq is the subgroup of those matrices with determinant 1. Describe the right cosets of SL2 pRq in GL2 pRq, and find the index of this subgroup. 4. Use Euler’s Theorem or Fermat’s Little Theorem to

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