homework3 - ϕ g Í = gg Í g-1 is an automorphism of G 7...

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Math 113 Section 5 Homework #3 Fall 2010 Due: Monday, September 27 1. (a) Show that the “orthogonal group” O ( n ) = { M GL ( n, R ) | det ( M ) = ± 1 } is indeed a group under matrix multiplication (b) Show that the “special linear group” SL ( n, R ) = { M GL ( n ) | det ( M ) = 1 } is indeed a group under matrix multiplication. 2. Determine whether the Klein 4 group Z / 2 Z × Z / 2 Z is isomorphic to the group Z / 4 Z . 3. Let the Heisenberg group H be the group of matrices of the form 1 x y 0 1 z 0 0 1 where x,y,z R . Show that every non-identity element of H has infinite order. 4. Let ϕ : G G Í be a homomorphism of groups. (a) Show that ker ( ϕ ) and im ( ϕ ) are groups. (b) Show that if g ker ( ϕ ) then for all g Í G , g Í g ( g Í ) - 1 ker ( ϕ ) . 5. Let ψ : Z / 6 Z Z / 8 Z be given by ψ a ) = ¯ a . Determine whether ψ is a group homomorphism. 6. Let G be a group and fix an element g G . Determine whether ϕ : G G defined by
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Unformatted text preview: ϕ ( g Í ) = gg Í g-1 is an automorphism of G . 7. Show that there are no isomorphisms between ( Q , +) and ( Q- { } , · ) . 8. Let G be a finite group with | G | = n and let H be one of its subgroups with | H | = m . (a) Fix an element g ∈ G and consider the set gH = { g Í ∈ G | g Í = gh for some h ∈ H } . Show that as g varies in G , the sets gH determine a partition of G . (b) Use your answer from (a) to show that m | n . 9. Let G be a group. Define its “center” as Z ( G ) = { g ∈ G | gg Í = g Í g for all g Í ∈ G } . (a) Show that Z ( G ) is a subgroup of G . (b) Let n ≥ 3 . Determine the center Z ( D 2 n ) of D 2 n , the dihedral group of order 2 n . 1...
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