Algebra 1 HW 2

Algebra 1 HW 2 - G = { e } . (b) If s G and g G , show that...

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Homework Assignment 2 Math 235 (Fall 2011) Due: Sept 28 (Wednesday) at the beginning of class (1) De±ne a binary operation on R by x y = x + y +1. Provethat( R , )isagroup . (2) Let G =GL 2 ( Z / 2 Z )beagroupw ithusua lmatr ixmu lt ip l icat ion . (a) Write down all elements in G and then ±nd the order of G . (b) Find the inverse of ° [0] [1] [1] [0] ± . (c) Find all g G so that g 2 = I ,where I is the 2 × 2ident itymatr ix . (3) Let G be a group and let g G . (a) Show that | g 1 | = | g | . (b) For h G ,showthat | hgh 1 | = | g | . (c) If | g | < , show that g 1 = g | g |− 1 . (4) Let G be a group and let G ° be the subgroup generated by { ghg 1 h
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Unformatted text preview: G = { e } . (b) If s G and g G , show that gsg 1 G . (5) Let G be an abelian group and let T be the set of all elements of G with nite order. (a) Show that T is a subgroup, called the torsion subgroup. (b) Show by example that T may not be a subgroup if G is not abelian. (Hint: In G = GL 2 ( R ), nd two nite order elements in G whose product is of innite order.) Department of Mathematics, McGill University, Montreal, QC, H3A 2K6 Canada E-mail address : hahn@math.mcgill.ca 1...
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