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MTH 461: Survey of Modern Algebra, Spring 2021 Homework 3 Homework 3 1. Let G be the group of symmetries of a square in the 2-dimensional plane. (a) Analogous to what we did for the symmetries of an equilateral triangle, write down all symmetries of the square. What is the order of the group G? (b) Label the 4 vertices of the square with the numbers 1, 2, 3, 4. Accounting for how these labels are moved around by each symmetry, write down a subgroup H Ă S4 which corresponds to G. (c) Is H all of S4 ? Is it contained in the alternating group A4 ? 2. Compute the following compositions of permutations. (a) p1345qp234q (b) p143qp23qp24q (c) p1354q100 3. (a) What is ordpσq for σ P Sn equal to a cycle of length l? Explain. (b) Recall that an arbitrary permutation σ P Sn can be written as σ “ σ1 σ2 ⋯σk where each σi is a cycle and they are all

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