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# hw13 - a Prove that congruence modulo H is an equivalence...

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Modern Algebra 1 Homework 6.2 Spring 2010 Due February 24 Exercise 3. Let f : A B be a bijection of sets. Prove that the function F : Sym( A ) Sym( B ) given by F ( g ) = f g f - 1 is a group isomorphism. Conclude that if A has size n then Sym( A ) = S n . Exercise 4. Let G be a group and let H be a subgroup. Given x, y G we say that x is congruent to y modulo H provided x - 1 y H . If x and y are congruent modulo H then we write x y (mod
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Unformatted text preview: a. Prove that congruence modulo H is an equivalence relation on G . b. Determine the equivalence class of x ∈ G . c. If H is the trivial subgroup, when are two elements of G congruent modulo H ? Exercise 5. Show that the distinct cosets of H = SL 2 ( R ) in G = GL 2 ( R ) are given by ± x 0 1 ² H with x ∈ R ....
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