1. 2. 3 and 4 Assume that G is a group not necessarily abelian. For all a G, let
ID: 2968854 • Letter: 1
Question
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Assume that G is a group not necessarily abelian. For all a G, let C G (a) = {x G : ax = xa}. Prove that and that C G (a) is a subgroup of G, for all What is C G (e)? Consider the additive group Mat(2, Z) of all 2 times 2-square matrices with integer as entries under the addition of matrices. Prove that the set of all diagonal matrices D form a subgroup of Mat(2, Z). Prove that for all groups G, Z(G = C G (a). Let G = GL(2,R). Prove that Z(G) consist in diagonal matrices with a 0 in R.Explanation / Answer
Take the matrix A = ((a b) (c d)) and suppose it commutes with everything.
Then take M = ((0 1)(1 0)) , then AM=MA, you'll end up with a=d and b=c here.
Then take M = ((0 0)(0 1)) , then AM=MA, you'll end up withb=0
So the matrix are of the form (( a 0) (0 a)) for any a!=0 (det(A)=a^2 != 0)
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