Algebra 1 HW 4

# Algebra 1 HW 4 - (b Show that G is isomorphic to either Z/p...

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Homework Assignment 4 Math 235 (Fall 2011) Due: Oct 28 (Friday) at the beginning of class For convenience, denote Z /n Z by Z /n (1) Let p be a positive prime and let U p be the unit group of Z /p .S h owt h a t U p is cyclic and thus U p = Z / ( p 1) . (2) Let G act on X and X ° (i.e. X and X ° are G -sets). A map ϕ : X X ° is called a homomorphism of G -sets if ϕ ( g · x )= g · ϕ ( x )fora l l x X and g G . (a) Prove that the stabilizer G ϕ ( x ) contains the stabilizer G x . (b) Prove that the orbit of x maps to the orbit of ϕ ( x ). (3) Let G be a ±nite p -group, where p is a positive prime. Show that G has nontrivial center. In other words Z ( G ) ° = { e } . (4) Let G be a group of order p
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Unformatted text preview: (b) Show that G is isomorphic to either Z /p 2 or Z /p × Z /p . (5) Let M 2 ( R ) be the set of 2 × 2 matrices with entries in R . Then M 2 ( R ) is a ring with the usual matrix addition and matrix multiplication. (a) Show that °± a b − b a ² : a, b ∈ R ³ is a subring of M 2 ( R ). (b) Show that the center of M 2 ( R ) is Z ( M 2 ( R )) = { λI 2 : λ ∈ R } where I 2 is the 2 × 2 identity matrix. Department of Mathematics, McGill University, Montreal, QC, H3A 2K6 Canada E-mail address : [email protected] 1...
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