1202. BASIC THEORY OF GROUPS(a)ForA2GL.n;R/andb2Rn, define the transformationTA;bWRn!RnbyTA;b.x/DAxCb. Show that the set of all suchtransformations forms a groupG.(b)Consider the set of matricesAb01;whereA2GL.n;R/andb2Rn, and where the0denotes a 1-by-nrow of zeros. Show that this is a subgroup of GL.nC1;R/,and that it is isomorphic to the group described in part (a).(c)Show that the mapTA;b7!Ais a homomorphism fromGtoGL.n;R/, and that the kernelKof this homomorphism is iso-morphic toRn, considered as an abelian group under vector ad-dition.2.4.21.LetGbe an abelian group. For any integern > 0show that themap'Wa7!anis a homomorphism fromGintoG. Characterize thekernel of'. Show that ifnis relatively prime to the order ofG, then'isan isomorphism; hence for each elementg2Gthere is a uniquea2Gsuch thatgDan.2.5. Cosets and Lagrange’s TheoremConsider the subgroupHD fe; .1 2/gS3. For each of the six elements
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