336A2SOLNS.pdf - PMATH 336 Due: June 9, 2020 at 9:00 AM...

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PMATH 336Due: June 9, 2020 at 9:00 AMAssignment 2 Comments1. (Xingchi) LetIbe the index set of nonzero real numbers and for everyαIletGαbe the group of nonzero complex numbers equipped with the operationa ? b=αab.ConsiderG=YαIGα.(a) What is the identity ofG?
(b) ConsiderfGgiven byf(α) =α+i. What isf-1?
(c) Letfbe as above and letgGbe given byg(α) = 1 +αi. Computefg.
2. (Aiden)(a) LetIbe a nonempty index set and for everyiIletGibe a group. SupposeHiGifor everyiI. Prove thatYiIHiYiIGi.1
PMATH 336Due: June 9, 2020 at 9:00 AMProof.Leteidenote the identity ofGi.First, the identityf(i) =eiHiis an element ofQHi.Now supposef, gQHiso thatf(i), g(i)HiforeveryiI. Then,f(i)g(i)-1Hifor everyiI, sinceHiGi. Thereforefg-1QHi, as required by the subgroup test.(b) Prove or Disprove: LetGbe a group. Every subgroup ofG×Gis of the formH×K, whereHGandKG.

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Term
Spring
Professor
MOSHEKAMENSKY
Tags
Cyclic group, F G

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