Solution Outlines for Chapter 7# 1: LetH={(1),(12)(34),(13)(24),(14)(23)}.Find the left cosets ofHinA4(using table 5.1).There are three cosets:H={(1),(12)(34),(13)(24),(14)(23)}={α1, α2, α3, α4}(123)H={(123),(134),(243),(142)}={α5, α6, α7, α8}(132)H={(132),(143),(234),(124)}={α9, α10, α11, α12}# 2: LetHbe as in Exercise 1.How many left cosets ofHinS4are there?(Determine this without listing them.)
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# 3: LetH={0,±3,±6,±9, . . .}. Find all the left cosets ofHinZ.
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# 5: LetHbe as in Exercise 3. Decide whether or not the following cosets ofHare the same.To do this, we use thataH=bHiffa-1b∈H, which in additive notation is (-a)+b∈H.
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# 8: Suppose thatahas order 15. Find all of the left cosets of< a5>in< a >.
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# 14: LetC*be the group of nonzero complex numbers under multiplication andletH={a+bi∈C*|a2+b2= 1}. Give a geometric description of the coset(3 + 4i)H. Give a geometric description of the coset(c+di)H.
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