Suppose that G is a group, and let S be a subset of G. We define...

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Suppose that G is a group, and let S be a subset of G. We define the centraliser of S in G (usually just "centraliser of S") to be CG(S) = {g G|gx=xg for all x € S}. In other words, Cc(S) is the set of all elements of G that commute with every element of S. 1. Show that for any S, CG(S) is a subgroup of G. 2. Show that if S₁ S2, then CG (S1) 2 CG (S₂). 3. Prove that CG(S) = CG((S)) (that is, the centraliser of S is the same as the centraliser of the subgroup generated by S). Because of part 3, people usually only think about the case where S is a subgroup. Also, taking S as big as possible (i.e. all of G) we get the set of elements of G that commute with all other elements of G. This is important enough that it gets its own name and symbol; the centre of G is Z(G) = Cc (G) = {g €G | gh=hg for all h € G}. 4. Check that in the case of G = GL₂(C), Z(G) consists of the nonzero "scalar" matrices: Z(GL2(C)) - {[5 2] 1²+0}. Hint: check under what conditions a matrix c d commutes with the following two elements of GL₂(C): 6181 Suppose that G is a group, and let S be a subset of G. We define the centraliser of S in G (usually just "centraliser of S") to be CG(S) = {g G|gx=xg for all x € S}. In other words, Cc(S) is the set of all elements of G that commute with every element of S. 1. Show that for any S, CG(S) is a subgroup of G. 2. Show that if S₁ S2, then CG (S1) 2 CG (S₂). 3. Prove that CG(S) = CG((S)) (that is, the centraliser of S is the same as the centraliser of the subgroup generated by S). Because of part 3, people usually only think about the case where S is a subgroup. Also, taking S as big as possible (i.e. all of G) we get the set of elements of G that commute with all other elements of G. This is important enough that it gets its own name and symbol; the centre of G is Z(G) = Cc (G) = {g €G | gh=hg for all h € G}. 4. Check that in the case of G = GL₂(C), Z(G) consists of the nonzero "scalar" matrices: Z(GL2(C)) - {[5 2] 1²+0}. Hint: check under what conditions a matrix c d commutes with the following two elements of GL₂(C): 6181

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