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HOMEWORK7

Algebra

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Math 150a: Modern Algebra Homework 7 This problem set is due Wednesday, November 14. Do problem 5.9.2 and the following problems: GK1. A review problem in set arithmetic. a. If X is a subset of a group G , then X 2 does not mean the same thing as XX : X 2 is the set of squares of elements in X , while XX is the set of products of pairs of elements. Show that if G is a finite group, then G 2 = G as sets if and only if G has an odd number of elements. (Hint: You can show that the squaring map is surjective if | G | is odd; and that it is not injective if | G | is even. Lagrange’s theorem and Cauchy’s theorem may be helpful.) b. If H is a normal subgroup of G , you may worry about interpreting an exponential ( aH ) n in the quotient group G / H , since ( aH ) n does not have to be a coset by part (a). Show that ( aH ) n is however contained in a unique coset of
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Unformatted text preview: H , namely a n H . (So we can modify set arithmetic on cosets using containment. This wrinkle comes up again in 150b when defining quotient rings.) GK2. As I will have said in class, if G is a group and H is a subgroup, then G and H are equiconjugate if they have this property: If a , b ∈ H are conjugate in G , then they are also conjugate in H . (Actually I made up this word; I do not know the official terminology.) Is the tetrahedral group T equiconjugate with the rotation group SO ( 3 ) ? Is the octahedral group O equiconjugate with SO ( 3 ) ? (The groups T and O are defined in section 5.9.) *GK3. How many isometries does an n-dimensional cube possess? (As a subgroup of O ( n ) , say.)...
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