College Algebra Exam Review 110

College Algebra - 120(a(b(c 2 BASIC THEORY OF GROUPS For A 2 GL.n R and b 2 Rn define the transformation TA;b W Rn Rn by TA;b.x D Ax C b Show that

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 120 (a) (b) (c) 2. BASIC THEORY OF GROUPS For A 2 GL.n; R/ and b 2 Rn , define the transformation TA;b W Rn ! Rn by TA;b .x / D Ax C b. Show that the set of all such transformations forms a group G . Consider the set of matrices Ä Ab ; 01 where A 2 GL.n; R/ and b 2 Rn , and where the 0 denotes a 1by-n row of zeros. Show that this is a subgroup of GL.n C 1; R/, and that it is isomorphic to the group described in part (a). Show that the map TA;b 7! A is a homomorphism from G to GL.n; R/, and that the kernel K of this homomorphism is isomorphic to Rn , considered as an abelian group under vector addition. 2.4.21. Let G be an abelian group. For any integer n > 0 show that the map ' W a 7! an is a homomorphism from G into G . Characterize the kernel of ' . Show that if n is relatively prime to the order of G , then ' is an isomorphism; hence for each element g 2 G there is a unique a 2 G such that g D an . 2.5. Cosets and Lagrange’s Theorem Consider the subgroup H D fe; .1 2/g  S3 . For each of the six elements 2 S3 you can compute the set H D f W 2 H g. For example, .23/H D f.23/; .132/g. Do the computation now, and check that you get the following results: eH D .1 2/H DH .2 3/H D .1 3 2/H D f.2 3/; .1 3 2/g .1 3/H D .1 2 3/H D f.1 3/; .1 2 3/g: As varies through S3 , only three different sets H are obtained, each occurring twice. Definition 2.5.1. Let H be subgroup of a group G . A subset of the form gH , where g 2 G , is called a left coset of H in G . A subset of the form H g, where g 2 G , is called a right coset of H in G . Example 2.5.2. S3 may be identified with a subgroup of S4 consisting of permutations that leave 4 fixed and permute f1; 2; 3g. For each of the 24 elements 2 S4 you can compute the set S3 . ...
View Full Document

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online