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College Algebra Exam Review 116

College Algebra Exam Review 116 - 126 2 BASIC THEORY OF...

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Unformatted text preview: 126 2. BASIC THEORY OF GROUPS not a linear space. So first explore the condition for a matrix A to commute with all matrices B , not just invertible matrices. Note that the condition AB D BA is linear in B , so a matrix A commutes with all matrices if, and only if, it commutes with each member of a linear spanning set of ma- trices. So now consider the so-called matrix units E ij , which have a 1 in the i;j position and zeros elsewhere. The set of matrix units is a basis of the linear space of matrices. Find the condition for a matrix to commute with all of the E ij ’s. It remains to show that if a matrix commutes with all invertible matrices, then it also commutes with all E ij ’s. (The results of this exercise hold just as well with the real numbers R replaced by the complex numbers C or the rational numbers Q .) 2.5.16. Show that the symmetric group S n has a unique subgroup of index 2, namely the subgroup A n of even permutations. Hint: Such subgroup N is normal. Hence if it contains one element of a certain cycle structure,is normal....
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