# Ch7Sltns.pdf - Solution Outlines for Chapter 7 # 1: Let H =...

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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.)
# 3: LetH={0,±3,±6,±9, . . .}. Find all the left cosets ofHinZ.
# 5: LetHbe as in Exercise 3. Decide whether or not the following cosets ofHare the same.To do this, we use thataH=bHiffa-1bH, which in additive notation is (-a)+bH.
# 8: Suppose thatahas order 15. Find all of the left cosets of< a5>in< a >.
# 14: LetC*be the group of nonzero complex numbers under multiplication andletH={a+biC*|a2+b2= 1}. Give a geometric description of the coset(3 + 4i)H. Give a geometric description of the coset(c+di)H.
25(a2+b2) = 25.In general, the coset (c+di)H, by similar computation, is the circle centered at the originwith radiusc2+d2.

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